Coding and modulation apparatus using non-uniform constellation

ABSTRACT

A coding and modulation apparatus and method are presented. The apparatus comprises an encoder that encodes input data into cell words, and a modulator that modulates said cell words into constellation values of a non-uniform constellation. The modulator is configured to use, based on the total number M of constellation points of the constellation and the code rate, a non-uniform constellation from one or several groups of constellations each comprising one or more constellations.

CROSS REFERENCE TO RELATED APPLICATIONS

This application is a continuation of and claims priority to U.S.application Ser. No. 17/119,678, filed Dec. 11, 2020, which is acontinuation of U.S. application Ser. No. 16/406,746, filed May 8, 2019(now U.S. Pat. No. 10,868,706, issued Dec. 15, 2020), which is acontinuation of U.S. application Ser. No. 16/105,666, filed Aug. 20,2018 (now U.S. Pat. No. 10,326,635, issued Jun. 18, 2019), which is acontinuation of U.S. application Ser. No. 15/699,635, filed Sep. 8, 2017(now U.S. Pat. No. 10,091,045, issued Oct. 2, 2018), which is acontinuation of U.S. application Ser. No. 15/340,672, filed Nov. 1, 2016(now U.S. Pat. No. 9,813,279, issued Nov. 7, 2017), which is acontinuation of U.S. application Ser. No. 14/901,894, filed Dec. 29,2015 (now abandoned), which is a National Stage of PCT/EP2014/064404,filed Jul. 7, 2014, and claims the benefit of priority to EuropeanPatent Application No. 14168129.6, filed May 13, 2014, European PatentApplication No. 14153438.8, filed Jan. 31, 2014, European PatentApplication No. 13183318.8, filed Sep. 6, 2013, and European PatentApplication No. 13175370.9, filed Jul. 5, 2013, the contents of each ofwhich are hereby incorporated by reference in their entirety.

BACKGROUND Field of the Disclosure

The present disclosure relates to a coding and modulation apparatus andmethod. Further, the present disclosure relates to a transmissionapparatus and method. Still further, the present disclosure relates to acomputer program and a non-transitory computer-readable recordingmedium.

Description of Related Art

Modern communications systems typically employ, among other elements, acoding and modulation apparatus (as part of a transmission apparatus)and a decoding and demodulation apparatus (as part of a receivingapparatus). The coding and modulation apparatus is often part of a socalled BICM (Bit Interleaved Coded Modulation) apparatus, whichgenerally comprises (at the transmitter side) a serial concatenation ofa FEC (Forward Error Correction) encoder, a bit interleaver, and amodulator, which uses spectral efficient modulation such as multilevelPAM (Pulse Amplitude Modulation), PSK (Phase Shift Keying), or QAM(Quadrature Amplitude Modulation). It should be noted that hereinafter,whenever QAM is mentioned it should be understood as a generally termcovering PAM, PSK and QAM.

BICM allows for good performance over both non-fading and fadingchannels due to the use of the interleaver and/or the FEC encoder. Ithas a reasonable decoding complexity as opposed to multilevel coding(MLC) coding schemes and is thus used frequently in communicationssystems, such as in all DVB systems, powerline communications (e.g.,Homeplug AV, DAB, LTE, WiFi, etc.).

Generally, the coding and modulation capacity, such as the BICM capacityin systems using a BICM apparatus, is considered as a target function,and it is desired to find optimum constellation points such that thiscapacity is maximized, often subject to a power normalization, i.e., theaverage power of the constellation points should be normalized to e.g.1.

The “background” description provided herein is for the purpose ofgenerally presenting the context of the disclosure. Work of thepresently named inventor(s), to the extent it is described in thisbackground section, as well as aspects of the description which may nototherwise qualify as prior art at the time of filing, are neitherexpressly or impliedly admitted as prior art against the presentdisclosure.

SUMMARY

It is an object to provide a coding and modulation apparatus and methodproviding an increased or even maximized capacity, a reduced bit errorrate and reception with a reduced SNR (signal-to-noise ratio). It is afurther object to provide a corresponding computer program and anon-transitory computer-readable recording medium for implementing saidmethods.

According to an aspect there is provided a coding and modulationapparatus comprising

-   -   an encoder that encodes input data into cell words, and    -   a modulator that modulates said cell words into constellation        values of a non-uniform constellation,        wherein said modulator is configured to use, based on the total        number M of constellation points of the constellation and the        code rate,

-   i) a non-uniform constellation from a group A or B of constellations    comprising one or more of the following constellations defined by    the constellation position vector u of length v=sqrt(M)/2−1,    -   wherein in one or more constellation position vectors of the        constellations from group B two or more constellation positions        are identical resulting from a condensation of preliminary        constellation positions optimized before,        or

-   ii) a non-uniform constellation from a group C or D of    constellations comprising one or more of the following    constellations, wherein the constellation points are defined by a    constellation position vector w_(0 . . . b-1) with b=M/4, wherein    -   a first quarter of the constellation points x_(0 . . . b-1) are        defined as x_(0 . . . b-1)=w_(0 . . . b-1),    -   a second quarter of the constellation points x_(b . . . 2b-1)        are defined as x_(b . . . 2b-1)=conj(w_(0 . . . b-1)),    -   a third quarter of the constellation points x_(2b . . . 3b-1)        are defined as x_(2b . . . 3b-1)=−conj (w_(0 . . . b-1)), and    -   a fourth quarter of the constellation points x_(3b . . . 4b-1)        are defined as x_(3b . . . 4b-1)=−w_(0 . . . b-1),    -   wherein conj is the complex conjugate,    -   wherein in one or more constellation position vectors of the        constellations from group D two or more constellation positions        are identical resulting from a condensation of preliminary        constellation positions optimized before,        or

-   iii) a non-uniform constellation from a group E of constellations    comprising one or more of the following constellations, wherein the    constellation points are defined by a constellation position vector    w_(0 . . . M-1),    or

-   iv) a non-uniform constellation obtained from a constellation from    anyone of groups A, B, C, D or E through rotation by an angle around    the origin, through inversion of bit labels for all constellation    points, through interchanging of bit positions and/or through    predistortion for the (in particular for all) constellation points,    wherein the constellation position vectors of the different    constellations of the groups A, B, C, D and E of constellations are    defined as follows, wherein the respective values R of the code rate    shall be understood as range of code rates R±1/30.

According to a further aspect there is provided a transmission apparatuscomprising

-   -   a coding and modulation apparatus as proposed herein that        encodes and modulates input data into constellation values,    -   a converter that converts said constellation values into one or        more transmission streams to be transmitted, and    -   a transmitter that transmits said one or more transmission        streams.

According to still further aspects corresponding methods, a computerprogram comprising program means for causing a computer to carry out thesteps of the coding and modulation method disclosed herein, when saidcomputer program is carried out on a computer, as well as anon-transitory computer-readable recording medium that stores therein acomputer program product, which, when executed by a processor, causesthe coding and modulation method disclosed herein to be performed areprovided.

Preferred embodiments are defined in the dependent claims. It shall beunderstood that the claimed methods, the claimed computer program andthe claimed computer-readable recording medium have similar and/oridentical preferred embodiments as the claimed apparatus and as definedin the dependent claims.

One of the aspects of the disclosure is that the constellation points ofthe used non-uniform constellations (herein also called NUCs) are notlocated on a regular grid with equidistant symbols, but rather onoptimized locations, dependent on the code rate of forward errorcorrection encoder, like an LDPC or a turbo code or any other known codeencoder (generally another forward error correction code encoder, e.g. aBCH encoder, may be provided in addition). Further, the usedconstellation may be selected (preferably in advance, but generally onthe fly in other embodiments) dependent on the desired total number ofconstellation points of the used constellation (and, in someembodiments, on the channel characteristics).

In the tables various constellations are provided for different valuesof M and for different code rates. It should be noted that the code rateR indicated in the tables are not to be understood such that aparticular constellation is only valid for exactly this code rate, butalso for slightly different code rates, i.e. a range of code ratesR±1/30. For instance, the indication 6/15 for a code rate (i.e. R=6/15)given in a proposed table means that the respective constellation isvalid for a range of code rates 6/15±1/30, i.e. for the range of codesrates from 11/30 to 13/30.

It should also be noted that one or more of the following “invarianttransformations” do not affect the properties of the constellations:

1. rotation of all symbols by an arbitrary angle φ,

2. inversion of m-th bit y_m=b∈{0,1} to y_m=⁻b, where the bar indicatesinversion,

3. interchanging of bit positions y_k1 and y_k2,

4. reflection on Re{x1}- and/or Im{x1}-axis,

5. predistortion (particularly for constellations of group E).

Thus, the modulator may also use a non-uniform constellation obtainedfrom a constellation from anyone of groups A, B, C, D or E throughrotation by an angle around the origin, an inversion of bit labels forall constellation points, an interchanging of bit positions for allconstellation points and/or a reflection on the real part and/orimaginary part axis. For instance, if one constellation point has bitlabels 0010 for 16-QAM, all first bit labels can be inverted such thatthis point becomes 1010. Further, constellation obtained through anyother trivial manipulation, such as rounding of the constellationpoints' positions shall generally be covered by the claims. Through oneor more of these operations an equivalent mapping to the mapping of theconstellations defined in the above mentioned four groups is achieved.

In case of satellite transmission, the modulator might as well transmitdifferent constellation points, obtained by predistortion of proposedconstellation points. This predistortion should can act as acountermeasure to the non-linearities of other blocks in thetransmission system, in particular the power amplifier. The output ofthe transmission system however, should correspond to the transmissionof the proposed constellations, such that the receiver might assume thatthese constellations have been transmitted.

It should be noted that to every M-QAM, one can also think of theunderlying sqrt(M)-PAM. Further, it should be noted that in otheraspects the group of constellations defined in the claims comprises lessconstellations, e.g. only constellations for non-fading channels, onlyconstellations for fading channels, only constellations for selectedvalues of M, only constellation for M-QAM or sqrt(M)-PAM and/orconstellations for less SNR values. In other words, less constellationsmay be contained in the group of constellations available for selectionand subsequent use by the modulator, i.e. the group of constellationsavailable for use by the modulator may comprise one or more of theconstellations defined in the claims. Accordingly, the presentdisclosure is also directed to a coding and modulation apparatus andmethod that have a smaller group of constellations available for use (asexplained above) and/or where less constellations are available for aparticular value of M.

A QAM mapping consisting of M constellation points is denoted as M-QAM.These constellations are summarized in group A. If a (uniform ornon-uniform) QAM allows separate encoding and decoding of each of itstwo dimensions (“inphase” and “quadrature phase” in the literature),then this QAM will be called a N²-QAM. This implies that theconstellation can be designed by two N-PAM constellations, one for eachdimension. N²-QAMs have significantly lower decoding complexity forML-decoding, as only N constellation points have to be investigated,compared with N² points for the M-QAM, when M=N², but when the twodimensions cannot be separated (as is usually the case for N-PSK, e.g.8-PSK, where 8 points are located on a unit circle). In addition QAMconstellations that are completely defined by a a quarter ofconstellation values of the constellation will be called QQAM, with theother three quarters of constellation values being derived from thefirst quarter. E.g. normal uniform square QAM constellations (UC) arealso QQAM constellations, due to their symmetry. These QQAMconstellations are summarized in group C.

However, the constellation points of the QAM constellations according toembodiments considered in this disclosure are not located on a regulargrid with equidistant symbols, but rather on optimized locations,dependent on the code rate.

According to the present disclosure an N²-NUC optimization based onN-PAM optimization is considered, combined with a dynamic reduction ofthe number of constellation points guaranteeing a well definedperformance with respect to the performance of the N²-NUC withoutreduction of the number of constellation points.

Constellation sizes up to 1024-QAM will be considered, where largeshaping gains are possible, especially in the high SNR region. By meansof a dynamic reduction (also called condensation in the following) ofconstellation points that are close to each other, the number ofconstellations points and, thus, the required storage and decodingcapacity can be significantly reduced. These constellations aresummarized in groups B and D. For example, the 1024-Q-QAM constellationoptimized for code rate 6/15 can be reduced from 1024 to 268constellation points without significant impact on the performance.

It should be noted that the constellation position vector w as definedin the claims directed to a preferred embodiment needs not necessarilycontain the constellation points of the first quarter of constellationpoints of the constellation, but could also contain the constellationpoints of any of the four quarters (expressed by the definition “of afirst quarter” in the claims). Due to the symmetry (which is somewhatprovided, but may not be readily visible by the bit labels; theconstellation points are generally symmetric with respect to thequadrants) this leads to constellations with a different bit mapping butwith identical performance. The constellation position vector w in thetables defined herein should therefore be considered as an example forall four symmetric constellations with different bit mapping butidentical performance.

It is to be understood that both the foregoing general description ofthe disclosure and the following detailed description are exemplary, butare not restrictive, of the disclosure.

BRIEF DESCRIPTION OF THE DRAWINGS

A more complete appreciation of the disclosure and many of the attendantadvantages thereof will be readily obtained as the same becomes betterunderstood by reference to the following detailed description whenconsidered in connection with the accompanying drawings, wherein:

FIG. 1 shows an embodiment of a coding and modulation apparatusaccording to the present disclosure,

FIG. 2 shows an embodiment of a transmission apparatus according to thepresent disclosure,

FIG. 3 shows an embodiment of a communications system according to thepresent disclosure,

FIG. 4 shows a regular 4-QAM constellation as a simple example for aconstellation,

FIG. 5A shows a 8-PAM non-uniform constellation,

FIG. 5B shows a 64-QAM non-uniform constellation,

FIG. 6 shows a constellation for a 64-QAM non-uniform constellationgenerally defining the constellation points,

FIG. 7 shows a non-uniform 16-QQAM constellation,

FIG. 8 shows a diagram illustrating the performance of non-uniformN²-QAM constellations,

FIG. 9 shows an example for 1D condensing according to an embodiment ofthe present disclosure,

FIG. 10 shows an example for 2D condensing according to an embodiment ofthe present disclosure,

FIG. 11A shows a non-uniform 1024-QAM constellation at 27 dB SNR,

FIG. 11B shows a non-uniform 1024-QAM constellation at 8 dB SNR,

FIG. 11C shows an enlargement of the N²-NUC optimized for 8 dB SNR.

FIG. 12A shows a non-uniform 64-QQAM constellation,

FIG. 12B shows a non-uniform 64-QQAM constellation,

FIG. 13 shows a diagram for a constellation from group E,

FIG. 14 shows a diagram illustrating the shaping gain for the aboveexplained non-uniform 8-ary QAM constellations, and

FIGS. 15AA-15CB show examples of LDPC codes for which the disclosedconstellations have been optimized.

DESCRIPTION OF THE EMBODIMENTS

Referring now to the drawings, wherein like reference numerals designateidentical or corresponding parts throughout the several views, FIG. 1shows an embodiment of a coding and modulation apparatus 10 according tothe present disclosure. It comprises an encoder 11 that encodes inputdata into cell words, and a modulator 12 that modulates said cell wordsinto constellation values of a non-uniform constellation. Said modulator12 is configured to use, based on the total number M of constellationpoints of the constellation and the code rate, a non-uniformconstellation from one of several groups of constellations. Details ofthose different groups of constellations will be explained in moredetail below.

In other embodiments of the coding and modulation apparatus 10additional elements may be provided, such as a BCH encoder, an LDPCencoder (whose code rate is of interest for selecting and using anoptimized constellation by the modulation disclosed herein), a bitinterleaver and/or a demultiplexer (for demultiplexing bits of encodeddata into the cell words). Some or all of these elements may separateelements or may be part of the encoder 11. For instance, a BICM deviceas conventionally used in the transmission apparatus of a DVB system maybe used as coding and modulation apparatus 10.

FIG. 2 shows an embodiment of a transmission apparatus 20 according tothe present disclosure comprising a coding and modulation apparatus 21(referenced by 10 in FIG. 1 ) as proposed herein that encodes andmodulates input data into constellation values, a converter 22 thatconverts said constellation values into one or more transmission streamsto be transmitted, and a transmitter 23 that transmits said one or moretransmission streams. In an exemplary embodiment the converter 22 maycomprise one or more elements like atime, cell and/or frequencyinterleaver, a frame builder, an OFDM modulator, etc., as e.g. describedin the various standards related to DVB and as may be provided in acoding and modulation apparatus in an upcoming ATSC standard. Othercoding and modulation in accordance with other standards, such as theDVB-S2 or subsequent DVB-Sx standards, may not comprises one or more ofthose elements. The constellations and the constellations values aregenerally predetermined and e.g. stored in a constellations storage 24or retrieved from an external source.

In other embodiments of the transmission apparatus 20 additionalelements may be provided, such as an input processing unit, a framebuilding unit and/or an OFDM generation unit as e.g. conventionally usedin a transmission apparatus of a DVB system.

FIG. 3 shows an embodiment of a communications system 30 according tothe present disclosure comprising one (or more) transmission apparatus20 (Tx) as shown in FIG. 2 and one or more receiving apparatus 40, 40′(Rx).

A receiving apparatus 40 generally comprises a receiver 41 that receivesone or more transmission streams, a deconverter 42 that deconverts thereceived one or more transmission streams into constellation values, anda demodulation and decoding apparatus 43 that demodulates and decodessaid constellation values into output data. The demodulation anddecoding apparatus 43 generally comprises a demodulator 44 fordemodulating constellation values of a non-uniform constellation intocell words, and a decoder 45 for decoding cell words into output datawords, wherein based on the total number M of constellation points ofthe constellation and the code rate, a non-uniform constellation isselected from the one of several groups of constellations comprising thesame predetermined constellations as used in the coding and modulationapparatus 10.

The preferred demodulation and decoding considers soft values as opposedto hard decided values (0 and 1). Soft values represent the continuouslydistributed received values (possibly after A/D conversion includingquantization) by more than two states (as in the case of binary (hard)decision). The reason is that for hard decision, the NUCs are generallynot optimal. Nowadays, BICM receivers typically are soft receiversanyway.

Generally, data (e.g. communications data, broadcast data, etc.) shallbe transmitted from a transmission apparatus 20 to one or more of saidreceiving apparatus 40 over a transmission channel 50, 50′. Thetransmission channel 50, 50′ can be unicast channel, multicast channel,a broadcast channel and may be employed as one-directional orbi-directional channel (i.e. having a return channel from the receivingapparatus to the transmission apparatus).

In an embodiment the modulator 12 is configured to use a non-uniformconstellation based on the total number M of constellation points of theconstellation, the required signal-to-noise ratio SNR for error freedecoding in dB and the channel characteristics. In broadcastingapplications the constellation is generally not selected dependent onthe SNR in the receiver, but dependent on the SNR that is required forerror free decoding with a used channel code (if a code is used, forexample LDPC codes in case of DVB 2^(nd) generation transmissionsystems) for an expected channel characteristic, e.g., static receptionor multipath fading.

For the broadcaster there may be a trade-off: using small constellationsizes M and/or low code rates R allow robust transmission (receptionalso at low SNR), but the throughput of the system scales withlog₂(M)*R. For instance, a 16-QAM with code rate R=3/4 can transmitthree information bits per coded QAM symbol. This results in relativelysmall spectral efficiency. On the other hand, high (spectral) efficiencyrequires large SNR. Thus, the constellations should allow decreasing theSNR required for successful decoding, while leaving the efficiencyconstant. This is the so called “shaping gain” of the optimizedconstellations.

The total number M of constellation points is generally selectedaccording to the desired payload throughput jointly with the code rateof the FEC encoder. The SNR for error free decoding for typical channelcharacteristic is generally known, e.g. by simulation. In broadcastingthe channel characteristics of the receivers are not known, i.e. acompromise is selected. For instance, in broadcasting for each code rateof the FEC encoder one non-uniform constellation is selected, optimizedfor an SNR that is a compromise for all channel characteristics.

The transmitter generally targets a certain scenario. For instance, abroadcast transmission over cable or satellite considers the channel tobe just a non-fading AWGN (appropriate channel model), while aterrestrial broadcaster typically considers the channel to be a fadingchannel, e.g. with Rayleigh distribution, as several echoes are usuallyreceived. Preferably, the proposed constellations consider the mostrelevant channel characteristics.

In another embodiment the modulator 12 is configured to adaptivelyselect a non-uniform constellation based on the total number M ofconstellation points of the constellation, the signal-to-noise ratio SNRin dB and the channel characteristics, wherein said signal-to-noiseratio SNR in dB and channel characteristics are received from areceiving device 40 to which data shall be transmitted. Such an adaptiveselection of the constellation is generally only possible with a returnchannel in unicast environments. A non-uniform constellation may beadapted e.g. in time and/or frequency domain, e.g. for different OFDMsubcarriers.

The channel characteristics describe the statistical properties of thechannel, e.g., the extent of the multipath propagation of thetransmission channel between transmitter and receiver. If the channel ischaracterized by no multipath propagation, corresponding to the AWGNchannel, the required SNR for error free decoding is relatively low,i.e. the NUC has to be selected accordingly for optimum performance. Ifthe transmission channel is characterized by strong multipathpropagation, the required SNR for error free reception is largercompared to a channel without multipath propagation, i.e. a NUCoptimized for higher SNR has to be used. Further, the NUCs should beoptimized taking the fading characteristics into account, as will bediscussed below.

As mentioned above, the number M of the constellation points of theconstellations is selected according to the desired payload throughput.Larger values of M allow for higher data throughput, but require alarger SNR for error free reception. This is further influenced by thecode rate of the FEC encoder, if any FEC encoder is used.

Another explanation (which is closely related to the optimization task)is that the performance of the constellation in combination with aforward error correction code (e.g. LDPC and/or BCH code) having acertain code rate shall be optimized. Thus, for various codes/code ratesoptimized constellations are proposed for different values of M. Anotheroptimization target is the capacity. For an expected SNR, say 15 dB ofSNR should be guaranteed, M is chosen, for which the respectiveoptimized NUC yields the largest capacity. As a general rule it holdsthat for low SNR a low value of M should be selected and vice versa. Butfrom a theoretical point of view, it turns out that high M is generallyoptimum, e.g., choosing M=4096 or M=1024 is preferred, because even forlow SNR, the optimized NUC will “look (almost) like” a constellationwith effectively smaller M, as several points will overlap. However,modulation and demodulation complexity increase with increasing M, so atradeoff is considered. Another aim is to reduce the BER (bit errorrate) and/or the FER (frame error rate) and/or to achieve the same BERand/or the FER at a lower SNR compared to a situation where a “normal”(not optimized) constellation is used.

A simple example of a constellation is shown in FIG. 4 . Here a regular4-QAM consisting of constellation points (e^(jπ/4), e^(j7π/4), e^(3π/4),e^(j5/4)) is depicted. The average symbol power is 1 (all symbols arelocated on unit circle here). The above symbol vector (e^(jπ/4),e^(j7π/4), e^(3π/4), e^(j5/4)) is to be understood such that the firstentry (e^(jπ/4)) belongs to the bit vector 00, the second entry(e^(j7π/4)) to 01 and so on, i.e. the entries belong to bit vectors withincreasing values, where the first bit position is the most significantbit (MSB) and the last one the least significant bit (LSB). This 4-QAMis a particular case of an N²-QAM, with N=2. Note that this definition(of being an N² QAM) does not only require N² being a square number(N²=2²), but also that the constellation is symmetrical and can bedescribed by two independent N-PAM constellations, here a 2-PAM: thein-phase component (real-part of the complex symbols) is a 2-PAM withsymbol vector (1/sqrt(2), −1/sqrt(2)) and describes the 1^(st) bit ofthe 4-QAM, whereas the quadrature-phase component (imaginary-part of thecomplex symbols) is the same 2-PAM, this time describing the 2^(nd) bitof the 4-QAM. Note further that the decomposition of the N²-QAM into twoN-PAMs is only possible if the bit labelling is according to binaryreflected Gray mapping, which is typically applied (e.g. inDVB-systems).

The above example can be extended to higher order N²-QAMs, with N>2.Then the underlying N-PAM describes for one component the 1^(st),3^(rd), 5^(th) and so on bit label, while for the other component itdescribes the 2^(nd), 4^(th), 6^(th) and so on label.

All constellations preferably fulfil power constraint, i.e

${\mathcal{P}_{x} = {{E_{x_{l}}\left\lbrack {x_{l}}^{2} \right\rbrack} = {{\frac{1}{M}{\sum\limits_{l = 0}^{M - 1}{x_{l}}^{2!}}} = 1}}},$where E[.] is the expectation operator, and x₁ is a particular symbol ofthe set of all M constellation symbols.

N²-NUCs have been optimized as one embodiment with N² being 16, 64, 256,1024 (1 k). This means that these constellations are optimized to allowminimum BER/FER for a given FEC code rate. The restriction on theseconstellations is that they should be able to be split into two separateone-dimensional PAM constellations, allowing low complexity mapping atthe transmitter and demapping at the receiver.

As an example, a M=64 NUC described here yield the following values (anexample from the tables could yield these three numbers, then there isthe 1 at the beginning (normalization due to power constraint) and soon):

2.27944.62297.5291.

This means that the positive constellation values are

12.27944.62297.5291

(the 1 was redundant, due to the power normalization, which will beapplied in the end). The underlying 1-dim. 8-PAM NUC is thus describedby the symbol vector(1.64051.00730.21790.4967-1.6405-1.0073-0.2179-0.4967), where the valuesare already normalized to unit average power.

As described before, the first entry (1.6405) corresponds to the bitlabel 000, the next one (1.0073) to 001 and so on. The 2-dim. 64-NUC isthen obtained by symmetry, where both in-phase and quadrature-phasecomponent of the NUC are based on the 8-PAM NUC.

FIG. 5A depicts 8-PAM NUC. FIG. 5B depicts 64-QAM NUC. The bit labelsare given in integer numbers (000→0,001→1,010→2 and so on).

The creation of the 2-dim. NUC based on the optimized degrees of freedomwill be explained in more detail below.

Since the performance of NUCs depends on the SNR value they areoptimized for, a thorough selection is preferably carried out dependingon the (FEC) code rate to achieve optimum performance. If the channelcharacteristics are known, the required SNR value for FEC convergencecan be determined by simulation. Then the NUC that has been optimizedfor this SNR value is chosen for best performance. If the SNR at thereceiver is lower than this SNR decoding threshold, the constellation isnot optimal. However, this is no drawback, since the capacity is too lowfor successful decoding anyhow. On the other hand if the SNR at thereceiver is clearly higher than the decoding threshold, a sufficientamount of capacity for successful decoding is available, even though theNUC is suboptimal for this SNR range. Therefore, the NUC needs to beoptimized for the SNR value at the waterfall region (i.e., decodingthreshold for (quasi-) error free decoding) of the FEC. As the SNR valueof the waterfall region depends on the code rate of the FEC, a differentNUC is selected for each code rate.

The SNR value for (quasi-) error free decoding also depends on thechannel characteristics of the receiver. For instance the required SNRfor error free decoding of the DVB-T2 LDPC code in the AWGN channel is0.8 dB, whereas 2.5 dB are required in the Rayleigh P1 multipathchannel. The selected NUC for each code rate is thus not optimal in allchannel environments and a tradeoff is necessary in a broadcastingenvironment that suits all (or most) users in the network. In apoint-to-point network with return channel, the optimal NUC may beselected based on the measured channel characteristics in the receiver.

In the following some more explanation is provided regarding thedefinition of the non-uniform QAM constellations. Each input cell word(y_(0,q) . . . y_(m-1,q)) (i.e. provided to the modulator) shall bemodulated using a non-uniform QAM constellation to give a constellationpoint z_(q) prior to normalization, where m corresponds to the number ofbits per QAM symbol m=log₂(M). It should be noted that the parameter qused here for discrete time or subcarrier index corresponds to theparameter k as used in the above. The exact values of the real andimaginary components Re(z_(q)) and Im(z_(q)) for each combination of therelevant input bits y_(0 . . . M-1,q) are given in the following tablesfor the various constellation sizes depending on the NUC position vectoru_(1 . . . v), which defines the constellation point position of thenon-uniform constellation. The length of the NUC position vector u isdefined by

$v = {\frac{\sqrt{M}}{2} - {1.}}$

In one example, the corresponding constellation point z_(q) for a 64-QAMNUC defined by the NUC position vector (u_(1 . . . 3))=(2,5,6) and theinput cell word (y_(0,q) . . . y_(m-1,q))=(100111) is Re(z_(q))=−u₂=−5and Im(z_(q))=u₁=2. The complete constellation for this NUC positionvector is shown in FIG. 6 with exemplary input cell words marked at thecorresponding constellation points.

The resulting constellation mapping (also called labeling) for thenon-uniform constellations follows a binary reflected Gray-Mapping(labeling), i.e. neighboring constellation points differ in only onebit. The power of the constellation points z_(q) is normalized such thatthe expectation value of the normalized constellation point f_(q) equals1, i.e. E(|f_(q)|²)=1. For example, the normalized constellation valuef_(q) of a uniform 16-QAM constellation results by

$f_{q} = {\frac{Z_{q}}{\sqrt{10}}.}$

The following tables define the constellation position vectors (prior topower normalization) as well as the bit labelling of the data cell wordsto the constellation points.

Constellation mapping for real part of 16-QAM y_(0,q) 1 1 0 0 y_(2,q) 01 1 0 Re(z_(q)) −3 −1 1 3 Uniform −u₁ −1 1 u₁ NUC

Constellation mapping for imaginary part of 16-QAM y_(1,q) 1 1 0 0y_(3,q) 0 1 1 0 Im(z_(q)) −3 −1 1 3 Uniform −u₁ −1 1 u₁ NUC

Constellation mapping for real part of 64-QAM y_(0,q) 1 1 1 1 0 0 0 0y_(2,q) 0 0 1 1 1 1 0 0 y_(4,q) 0 1 1 0 0 1 1 0 Re(z_(q)) −7 −5 −3 −1 13 5 7 Uniform −u₃ −u₂ −u₁ −1 1 u₁ u₂ u₃ NUC

Constellation mapping for imaginary part of 64-QAM y_(1,q) 1 1 1 1 0 0 00 y_(3,q) 0 0 1 1 1 1 0 0 y_(5,q) 0 1 1 0 0 1 1 0 Im(z_(q)) −7 −5 −3 −11 3 5 7 Uniform −u₃ −u₂ −u₁ −1 1 u₁ u₂ u₃ NUC

Constellation mapping for real part of 256-QAM y_(0,q) 1 1 1 1 1 1 1 1 00 0 0 0 0 0 0 y_(2,q) 0 0 0 0 1 1 1 1 1 1 1 1 0 0 0 0 y_(4,q) 0 0 1 1 11 0 0 0 0 1 1 1 1 0 0 y_(6,q) 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 Re(z_(q))−15 −13 −11 −9 −7 −5 −3 −1 1 3 5 7 9 11 13 15 Uniform −u₇ −u₆ −u₅ −u₄−u₃ −u₂ −u₁ −1 1 u₁ u₂ u₃ u₄ u₅ u₆ u₇ NUC

Constellation mapping for real part of 256-QAM y_(1,q) 1 1 1 1 1 1 1 1 00 0 0 0 0 0 0 y_(3,q) 0 0 0 0 1 1 1 1 1 1 1 1 0 0 0 0 y_(5,q) 0 0 1 1 11 0 0 0 0 1 1 1 1 0 0 y_(7,q) 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 Im(z_(q))−15 −13 −11 −9 −7 −5 −3 −1 1 3 5 7 9 11 13 15 Uniform −u₇ −u₆ −u₅ −u₄−u₃ −u₂ −u₁ −1 1 u₁ u₂ u₃ u₄ u₅ u₆ u₇ NUC

Constellation mapping for real part of 1024-QAM Y_(0,q) 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 y_(2,q) 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 y_(4,q) 0 0 0 01 1 1 1 1 1 1 1 0 0 0 0 y_(6,q) 0 0 1 1 1 1 0 0 0 0 1 1 1 1 0 0 y_(8,q)0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 Re(z_(q)) −31 −29 −27 −25 −23 −21 −19−17 −15 −13 −11 −9 −7 −5 −3 −1 Uniform −u₁₅ −u₁₄ −u₁₃ −u₁₂ −u₁₁ −u₁₀ −u₉−u₈ −u₇ −u₆ −u₅ −u₄ −u₃ −u₂ −u₁ −1 NUC Y_(0,q) 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 y_(2,q) 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 y_(4,q) 0 0 0 0 1 1 1 1 11 1 1 0 0 0 0 y_(6,q) 0 0 1 1 1 1 0 0 0 0 1 1 1 1 0 0 y_(8,q) 0 1 1 0 01 1 0 0 1 1 0 0 1 1 0 Re(z_(q)) 1 3 5 7 9 11 13 15 17 19 21 23 25 27 2931 Uniform 1 u₁ u₂ u₃ u₄ u₅ u₆ u₇ u₈ u₉ u₁₀ u₁₁ u₁₂ u₁₃ u₁₄ u₁₅ NUC

Constellation mapping for imaginary part of 1024-QAM Y_(1,q) 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 y_(3,q) 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 y_(5,q) 0 00 0 1 1 1 1 1 1 1 1 0 0 0 0 y_(7,q) 0 0 1 1 1 1 0 0 0 0 1 1 1 1 0 0y_(9,q) 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 Im(z_(q)) −31 −29 −27 −25 −23−21 −19 −17 −15 −13 −11 −9 −7 −5 −3 −1 Uniform −u₁₅ −u₁₄ −u₁₃ −u₁₂ −u₁₁−u₁₀ −u₉ −u₈ −u₇ −u₆ −u₅ −u₄ −u₃ −u₂ −u₁ −1 NUC Y_(1,q) 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 y_(3,q) 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 y_(5,q) 0 0 0 01 1 1 1 1 1 1 1 0 0 0 0 y_(7,q) 0 0 1 1 1 1 0 0 0 0 1 1 1 1 0 0 y_(9,q)0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 Im(z_(q)) 1 3 5 7 9 11 13 15 17 19 21 2325 27 29 31 Uniform 1 u₁ u₂ u₃ u₄ u₅ u₆ u₇ u₈ u₉ u₁₀ u₁₁ u₁₂ u₁₃ u₁₄ u₁₅NUC

Constellation mapping for real part of 4096-QAM Y_(0,q) 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 y_(2,q) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 y_(4,q) 0 0 0 00 0 0 0 1 1 1 1 1 1 1 1 y_(6,q) 0 0 0 0 1 1 1 1 1 1 1 1 0 0 0 0 y_(8,q)0 0 1 1 1 1 0 0 0 0 1 1 1 1 0 0 y_(10,q) 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0Re(z_(q)) −63 −61 −59 −57 −55 −53 −51 −49 −47 −45 −43 −41 −39 −37 −35−33 Uniform −u₃₁ −u₃₀ −u₂₉ −u₂₈ −u₂₇ −u₂₆ −u₂₅ −u₂₄ −u₂₃ −u₂₂ −u₂₁ −u₂₀−u₁₉ −u₁₈ −u₁₇ −u₁₆ NUC Y_(0,q) 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 y_(2,q)1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 y_(4,q) 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0y_(6,q) 0 0 0 0 1 1 1 1 1 1 1 1 0 0 0 0 y_(8,q) 0 0 1 1 1 1 0 0 0 0 1 11 1 0 0 y_(10,q) 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 Re(z_(q)) −31 −29 −27−25 −23 −21 −19 −17 −15 −13 −11 −9 −7 −5 −3 −1 Uniform −u₁₅ −u₁₄ −u₁₃−u₁₂ −u₁₁ −u₁₀ −u₉ −u₈ −u₇ −u₆ −u₅ −u₄ −u₃ −u₂ −u₁ −1 NUC Y_(0,q) 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 y_(2,q) 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1y_(4,q) 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 y_(6,q) 0 0 0 0 1 1 1 1 1 1 1 10 0 0 0 y_(8,q) 0 0 1 1 1 1 0 0 0 0 1 1 1 1 0 0 y_(10,q) 0 1 1 0 0 1 1 00 1 1 0 0 1 1 0 Re(z_(q)) 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31Uniform 1 u₁ u₂ u₃ u₄ u₅ u₆ u₇ u₈ u₉ u₁₀ u₁₁ u₁₂ u₁₃ u₁₄ u₁₅ NUC Y_(0,q)0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 y_(2,q) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0y_(4,q) 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 y_(6,q) 0 0 0 0 1 1 1 1 1 1 1 10 0 0 0 y_(8,q) 0 0 1 1 1 1 0 0 0 0 1 1 1 1 0 0 y_(10,q) 0 1 1 0 0 1 1 00 1 1 0 0 1 1 0 Re(z_(q)) 33 35 37 39 41 43 45 47 49 51 53 55 57 59 6163 Uniform u₁₆ u₁₇ u₁₈ u₁₉ u₂₀ u₂₁ u₂₂ u₂₃ u₂₄ u₂₅ u₂₆ u₂₇ u₂₈ u₂₉ u₃₀u₃₁ NUC

Constellation mapping for imaginary part of 4096-QAM y_(1,q)  1  1  1  1 1  1  1  1  1  1  1  1  1  1  1  1 y_(3,q)  0  0  0  0  0  0  0  0  0 0  0  0  0  0  0  0 y_(5,q)  0  0  0  0  0  0  0  0  1  1  1  1  1  1 1  1 y_(7,q)  0  0  0  0  1  1  1  1  1  1  1  1  0  0  0  0 y_(9,q)  0 0  1  1  1  1  0  0  0  0  1  1  1  1  0  0 y_(11,q)  0  1  1  0  0  1 1  0  0  1  1  0  0  1  1  0 Im(z_(q)) −63 −61 −59 −57 −55 −53 −51 −49−47 −45 −43 −41 −39 −37 −35 −33 Uniform −u₃₁ −u₃₀ −u₂₉ −u₂₈ −u₂₇ −u₂₆−u₂₅ −u₂₄ −u₂₃ −u₂₂ −u₂₁ −u₂₀ −u₁₉ −u₁₈ −u₁₇ −u₁₆ NUC y_(1,q)  1  1  1 1  1  1  1  1  1  1  1  1  1  1  1  1 y_(3,q)  1  1  1  1  1  1  1  1 1  1  1  1  1  1  1  1 y_(5,q)  1  1  1  1  1  1  1  1  0  0  0  0  0 0  0  0 y_(7,q)  0  0  0  0  1  1  1  1  1  1  1  1  0  0  0  0 y_(9,q) 0  0  1  1  1  1  0  0  0  0  1  1  1  1  0  0 y_(11,q)  0  1  1  0  0 1  1  0  0  1  1  0  0  1  1  0 Im(z_(q)) −31 −29 −27 −25 −23 −21 −19−17 −15 −13 −11  −9  −7  −5  −3  −1 Uniform −u₁₅ −u₁₄ −u₁₃ −u₁₂ −u₁₁−u₁₀ −u₉ −u₈ −u₇ −u₆ −u₅ −u₄ −u₃ −u₂ −u₁  −1 NUC y_(1,q)  0  0  0  0  0 0  0  0  0  0  0  0  0  0  0  0 y_(3,q)  1  1  1  1  1  1  1  1  1  1 1  1  1  1  1  1 y_(5,q)  0  0  0  0  0  0  0  0  1  1  1  1  1  1  1 1 y_(7,q)  0  0  0  0  1  1  1  1  1  1  1  1  0  0  0  0 y_(9,q)  0  0 1  1  1  1  0  0  0  0  1  1  1  1  0  0 y_(11,q)  0  1  1  0  0  1  1 0  0  1  1  0  0  1  1  0 Im(z_(q))  1  3  5  7  9  11  13  15  17  19 21  23  25  27  29  31 Uniform  1 u₁ u₂ u₃ u₄ u₅ u₆ u₇ u₈ u₉ u₁₀ u₁₁u₁₂ u₁₃ u₁₄ u₁₅ NUC y_(1,q)  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 0 y_(3,q)  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 y_(5,q)  1  1 1  1  1  1  1  1  0  0  0  0  0  0  0  0 y_(7,q)  0  0  0  0  1  1  1 1  1  1  1  1  0  0  0  0 y_(9,q)  0  0  1  1  1  1  0  0  0  0  1  1 1  1  0  0 y_(11,q)  0  1  1  0  0  1  1  0  0  1  1  0  0  1  1  0Im(z_(q))  33  35  37  39  41  43  45  47  49  51  53  55  57  59  61 63 Uniform u₁₆ u₁₇ u₁₈ u₁₉ u₂₀ u₂₁ u₂₂ u₂₃ u₂₄ u₂₅ u₂₆ u₂₇ u₂₈ u₂₉ u₃₀u₃₁ NUC

In an embodiment the modulator of the disclosed coding and modulationapparatus modulates said cell words into constellation values of anon-uniform constellation wherein said modulator is configured to use,based on the total number M of constellation points of the constellationand the code rate, a non-uniform constellation from a group A ofconstellations comprising one or more of the following constellationsdefined by the constellation position vector u of length v=sqrt(M)/2−1.

The following non-uniform constellations comprised in group A areproposed:

A) M-QAM Non-Uniform Constellations of Group A:

A1) 16-QAM NUC u/code rate 6/15 7/15 8/15 9/15 10/15 11/15 12/15 13/15u1 3.1169 3.1973 3.2334 3.2473 3.2436 3.2312 3.2149 3.1976

A2) 64-QAM NUC u/code rate 6/15 7/15 8/15 9/15 10/15 11/15 12/15 13/15u1 1.5409 2.1208 2.3945 2.6067 2.8505 2.912 2.9751 3.0032 u2 3.58264.3237 4.6577 4.9074 5.1674 5.2201 5.2491 5.2489 u3 5.5069 6.8108 7.34757.7177 8.0398 8.068 8.0217 7.9528

A3) 256-QAM NUC u/code rate 6/15 7/15 8/15 9/15 10/15 11/15 12/15 13/15u1 0.9918 0.9989  1.1155  1.3963  2.2282  2.6619  2.8437  2.9176 u22.2615 2.6086  2.8419  3.1795  4.1541  4.664   4.8758  4.956  u3 2.28732.7307  3.2659  3.9675  5.676   6.5386  6.8857  7.0096 u4 4.2761 4.6692 5.1383  5.9281  7.9072  8.8521  9.1906  9.282  u5 4.6871 5.3576  6.3082 7.4353 10.0292 11.2248 11.6157 11.6881 u6 6.5483 7.3828  8.4196  9.782512.8864 14.2018 14.5326 14.508  u7 8.6107 9.7612 11.0879 12.7927 16.563217.9894 18.1926 17.9984

A4) 1024-QAM NUC u/code rate 5/15 6/15 7/15 8/15 9/15 10/15 11/15 12/1513/15 u1 0.9997 1.0003 0.9994 1.0005  1.0023  1.0772  2.0789  2.5983 2.8638 u2 0.9916 1.0149 1.2742 2.0897  2.5667  2.8011  3.9147  4.5193 4.8422 u3 0.9911 1.0158 1.2749 2.0888  2.5683  2.9634  5.0664  6.1649 6.7392 u4 2.4349 2.6848 3.0323 3.9945  4.5468  4.8127  7.0579  8.2107 8.7961 u5 2.4346 2.6903 3.0371 3.9931  4.5636  5.1864  8.3596  9.959410.7659 u6 2.486  2.882  3.6813 5.3843  6.2876  6.7838 10.2901 12.032112.8844 u7 2.4864 2.8747 3.6718 5.3894  6.4073  7.5029 11.834  13.957414.98   u8 4.4576 4.7815 5.5854 7.5206  8.4282  9.238  14.0092 16.259817.2736 u9 4.4646 4.7619 5.5804 7.6013  8.8692 10.32   15.8419 18.426919.5552 u10 4.9706 5.5779 6.8559 9.3371 10.6112 12.0115 18.1472 20.927322.0472 u11 4.9552 5.6434 7.0475 9.8429 11.6946 13.5356 20.4243 23.486324.6335 u12 6.7222 7.3854 8.8436 11.9255  13.7334 15.6099 23.238126.4823 27.5337 u13 7.0327 7.8797 9.7042 13.3962  15.6274 17.752426.2322 29.7085 30.6651 u14 8.5382 9.635  11.727   15.8981  18.293320.5256 29.973  33.6247 34.3579 u15 10.4411  11.7874  14.2894  19.1591 21.7769 24.1254 34.7629 38.5854 38.9603

A5) 4096-QAM NUC u/code rate 5/15 6/15 7/15 8/15 9/15 10/15 11/15 12/1513/15 u1 1.0004 0.9998 0.9988 0.9999 0.9999 1.0009  1.0043  2.031  2.7135 u2 1.0009 0.9984 0.9996 1.0004 1.0364 2.272   2.7379  3.8448 4.6426 u3 1.0008 0.9983 0.998  1.0013 1.037  2.2732  2.746   4.8763 6.3617 u4 0.9953 1.2356 2.0374 2.5341 2.7604 4.1332  4.6828  6.8402 8.3598 u5 0.9956 1.2354 2.0374 2.5347 2.7604 4.1327  4.7079  7.912 10.0999 u6 0.9956 1.2372 2.0374 2.5328 2.8536 5.4651  6.4365  9.737312.0627 u7 0.9957 1.2366 2.0366 2.5362 2.8534 5.4655  6.5065 10.885913.8413 u8 2.5769 2.9864 3.93   4.5077 4.7342 7.4777  8.4445 12.889115.9032 u9 2.5774 2.9867 3.931  4.5089 4.7345 7.478   8.6065 14.109117.7263 u10 2.5794 2.9906 3.9297 4.5202 5.0028 8.9382 10.254  15.970419.7625 u11 2.5793 2.9912 3.9294 4.5204 5.0025 8.9384 10.5627 17.279721.6431 u12 2.6756 3.5858 5.277  6.2318 6.6184 10.9166  12.3025 19.294823.7857 u13 2.6764 3.5859 5.2757 6.2314 6.6189 10.9185  12.8281 20.718525.7402 u14 2.6747 3.577  5.2792 6.3219 7.215  12.5855  14.3539 22.672727.9051 u15 2.6746 3.5775 5.2795 6.3212 7.2149 12.5908  15.1126 24.243929.9595 u16 4.6252 5.474  7.4011 8.3718 8.9581 14.7944  16.7817 26.391632.2702 u17 4.6243 5.4757 7.3997 8.372  8.9585 14.8184  17.7277 28.106234.4336 u18 4.6188 5.4675 7.4646 8.7429 9.94   16.6805  19.2664 30.233536.7993 u19 4.62   5.4681 7.4665 8.7398 9.9394 16.8025  20.3968 32.124239.1019 u20 5.246  6.7028 9.1828 10.5001  11.5978  18.9131  22.100234.4644 41.6513 u21 5.2441 6.7104 9.1919 10.4999  11.6041  19.325 23.4306 36.5679 44.1297 u22 5.2556 6.8816 9.641  11.4966  12.9965 21.168  25.1704 39.027  46.8305 u23 5.2584 6.8738 9.6278 11.5018 13.088   22.0945 26.7556 41.4123 49.5435 u24 7.0279 8.6613 11.7039 13.483   14.878   24.1425 28.7536 44.2002 52.5353 u25 7.0459 8.634411.6913  13.5672  15.2586  25.575  30.6214 46.9287 55.5551 u26 7.43059.4101 12.9619  15.1187  16.8096  27.6541 32.829  50.0315 58.8707 u277.3941 9.5027 13.2128  15.6519  17.8237  29.6567 35.1146 53.2678 62.3471u28 8.8516 11.1654  15.161   17.6098  19.6965  32.2679 37.8272 57.008566.2558 u29 9.1461 11.7322  16.204   19.1046  21.4926  35.0526 40.768561.0847 70.5087 u30 10.4828  13.5243  18.4804  21.5413  23.9997  38.602344.3725 65.9903 75.5397 u31 12.3176  15.7967  21.4433  24.7641  27.2995 43.2007 48.9596 72.1993 81.8379

In the following the Q-NUC optimization will be described, i.e. theoptimization of a 2-dimensional constellation that is derived from asingle quadrant. The above described optimization of a N²-QAM requiresthe optimization of sqrt(M)/2−1 degrees of freedom. Since theoptimization of a 2-dimensional QAM constellation has 2*M degrees offreedom (real and imaginary part of each constellation point) theoptimization is significantly more time consuming. Since the optimum2D-constellations for the 16-QAM case are symmetric with respect to thedifferent quadrants of the constellations, the following simplificationscan be applied to describe these constellations: Only a first quarter ofthe total number of constellation points of a constellation (e.g. thefirst quadrant of the constellation) is described, reducing the numberof table entries from 2*M to M/2. From the first quarter the remainingquarters can be derived, leading to a so called QQAM constellation.However, it shall be ensured that the properties of the bit labeling ofthe constellation points are retained. For instance, if the firstquadrant is Gray-Mapped, offering a Hamming distance of 1 of the bitlabels belonging to adjacent constellation points, the same must beensured for the remaining quadrants of the QQAM constellation.

To uniquely define a 16-QQAM only 8 real values are required,corresponding to 4 complex values representing the constellation pointsof the first quadrant. Based on the QQAM approach 16-QQAM, 32-QQAM, 64QQAM, 128-QQAM, 256-QQAM and 1024-QQAM constellations have beenoptimized, clearly outperforming the N²-QAM constellations. Thepresented QQAM optimization approach can be used for any channelcondition, e.g. for the AWGN channel as well as for fading channels.

For other systems like a satellite communication system in accordancewith the DVB-S2 or Sx standard, constellations for M=8 constellationpoints have been optimized. These constellations cannot be described byonly a quarter of the constellations points. Rather all 8 complex valuesare explicitly described.

In an embodiment the modulator of the disclosed coding and modulationapparatus modulates said cell words into constellation values of anon-uniform constellation wherein said modulator is configured to use,based on the total number M of constellation points of the constellationand the code rate, a non-uniform constellation from a group C or D ofconstellations comprising one or more of the following constellations,wherein the constellation points are defined by a constellation positionvector w_(0 . . . b-1) with b=M/4, wherein

a first quarter of the constellation points x_(0 . . . b-1) are definedas x_(0 . . . b-1)=w_(0 . . . b-1),

a second quarter of the constellation points x_(b . . . 2b-1) aredefined as x_(b . . . 2b-1)=conj(w_(0 . . . b-1)),

a third quarter of the constellation points x_(2b . . . 3b-1) aredefined as x_(2b . . . 3b-1)=−conj (w_(0 . . . b-1)), and

a fourth quarter of the constellation points x_(3b . . . 4b-1) aredefined as x_(3b . . . 4b-1)=−w_(0 . . . b-1),

wherein conj is the complex conjugate.

The following non-uniform constellations comprised in group C areproposed (i=srqt(−1) is the imaginary unit):

C) M-QAM Non-Uniform Constellations of Group C:

C1) 16-QQAM NUC w/code rate 5/15 6/15 7/15 8/15 9/15 w0 1.1928 + 0.5309i0.2663 + 0.4530i 0.2626 + 0.4634i 0.2575 + 0.4819i 0.2474 + 0.5061i w10.5309 + 1.1928i 0.4530 + 0.2663i 0.4634 + 0.2626i 0.4819 + 0.2575i0.5061 + 0.2474i w2 0.4633 + 0.2842i 0.5115 + 1.2092i 0.5023 + 1.2100i0.4951 + 1.2068i 0.4909 + 1.2007i w3 0.2842 + 0.4633i 1.2092 + 0.5115i1.2100 + 0.5023i 1.2068 + 0.4951i 1.2007 + 0.4909i w/code rate 10/1511/15 12/15 13/15 w0 0.6186 + 0.2544i 0.2881 + 0.9576i −0.2949 + 0.2949i−0.2999 + 0.2999i w1 0.2213 + 0.4416i 0.2881 + 0.2881i −0.9555 + 0.2949i−0.9540 + 0.2999i w2 1.2080 + 0.5377i 0.9576 + 0.9576i −0.2949 + 0.9555i−0.2999 + 0.9540i w3 0.4487 + 1.1657i 0.9576 + 0.2881i −0.9555 + 0.9555i−0.9540 + 0.9540i

C2) 64-QQAM NUC w/code rate 5/15 6/15 7/15 8/15 9/15 w0 1.0257 + 0.5960i0.5656 + 0.9499i 0.2925 + 1.4892i 0.2920 + 1.4827i 0.2978 + 1.4669i w11.2181 + 0.7476i 0.2750 + 1.0676i 0.8449 + 1.2622i 0.8411 + 1.2563i0.8421 + 1.2355i w2 1.1509 + 0.3069i 0.8202 + 1.2651i 0.2351 + 1.0196i0.2174 + 1.0211i 0.2135 + 1.0389i w3 1.3888 + 0.3325i 0.3011 + 1.4529i0.5555 + 0.8926i 0.5702 + 0.8798i 0.6055 + 0.8654i w4 0.5961 + 1.0257i0.9500 + 0.5641i 1.4892 + 0.2925i 1.4827 + 0.2920i 1.4685 + 0.2859i w50.7476 + 1.2181i 1.0666 + 0.2744i 1.2622 + 0.8449i 1.2563 + 0.8410i1.2516 + 0.8201i w6 0.3069 + 1.1510i 1.2657 + 0.8178i 1.0196 + 0.2351i1.0211 + 0.2174i 1.0279 + 0.1981i w7 0.3325 + 1.3888i 1.4521 + 0.3005i0.8926 + 0.5555i 0.8798 + 0.5702i 0.8857 + 0.5642i w8 0.4969 + 0.2870i0.3004 + 0.5417i 0.1558 + 0.3029i 0.1475 + 0.3040i 0.1338 + 0.3767i w90.4161 + 0.2486i 0.2430 + 0.5607i 0.1712 + 0.3021i 0.1691 + 0.3028i0.1752 + 0.3563i w10 0.4969 + 0.2500i 0.2094 + 0.3579i 0.2075 + 0.6586i0.1871 + 0.6855i 0.1756 + 0.7261i w11 0.4084 + 0.2266i 0.1946 + 0.3566i0.3354 + 0.6030i 0.3563 + 0.6126i 0.4023 + 0.6180i w12 0.2870 + 0.4969i0.5410 + 0.3002i 0.3029 + 0.1558i 0.3040 + 0.1475i 0.2713 + 0.1337i w130.2486 + 0.4162i 0.5596 + 0.2431i 0.3021 + 0.1712i 0.3028 + 0.1691i0.2748 + 0.1572i w14 0.2500 + 0.4969i 0.3569 + 0.2094i 0.6586 + 0.2075i0.6855 + 0.1871i 0.6840 + 0.1578i w15 0.2266 + 0.4084i 0.3553 + 0.1948i0.6030 + 0.3354i 0.6126 + 0.3563i 0.6145 + 0.3556i w/code rate 10/1511/15 12/15 13/15 w0 0.2878 + 1.4388i −0.4661 + 0.9856i 1.0519 − 0.5188i1.0854 − 0.5394i w1 0.8133 + 1.2150i −0.4329 + 0.6825i 0.7146 − 0.4532i0.7353 − 0.4623i w2 0.2219 + 1.0386i −0.1534 + 1.0366i 1.0500 − 0.1642i1.0474 − 0.1695i w3 0.6145 + 0.8494i −0.1340 + 0.6796i 0.7170 − 0.1473i0.7243 − 0.1504i w4 1.4656 + 0.2931i −0.8178 + 1.1580i 1.0952 − 0.9115i1.0693 − 0.9408i w5 1.2278 + 0.8230i −0.8132 + 0.6913i 0.6868 − 0.8108i0.7092 − 0.8073i w6 1.0649 + 0.2069i −0.2702 + 1.4529i 1.4480 − 0.2403i1.4261 − 0.2216i w7 0.8971 + 0.5677i −1.2125 + 0.7097i 0.6406 − 1.1995i0.6106 − 1.1783i w8 0.1177 + 0.4119i −0.4145 + 0.1264i 0.1325 − 0.3998i0.1392 − 0.4078i w9 0.2516 + 0.3998i −0.4179 + 0.3948i 0.4122 − 0.4120i0.4262 − 0.4205i w10 0.1559 + 0.7442i −0.1353 + 0.1272i 0.1374 − 0.1295i0.1407 − 0.1336i w11 0.4328 + 0.5954i −0.1359 + 0.3877i 0.4185 − 0.1357i0.4265 − 0.1388i w12 0.1678 + 0.1166i −0.7330 + 0.1416i 0.1369 − 0.7073i0.1388 − 0.7057i w13 0.3325 + 0.1582i −0.7177 + 0.4018i 0.4044 − 0.7057i0.4197 − 0.7206i w14 0.7408 + 0.1355i −1.0718 + 0.1686i 0.1677 − 1.0405i0.1682 − 1.0316i w15 0.6200 + 0.3227i −1.4375 + 0.2732i 0.2402 − 1.4087i0.2287 − 1.3914i

C3) 256-QQAM NUC w/code rate 5/15 6/15 7/15 8/15 9/15 w0 −0.3495 +1.7549i 0.6800 + 1.6926i 0.7280 + 1.6384i 1.0804 + 1.3788i 1.6350 +0.1593i w1 −0.2804 + 1.4293i 0.3911 + 1.3645i 0.4787 + 1.3492i 1.0487 +0.9862i 1.5776 + 0.4735i w2 −0.2804 + 1.4293i 0.2191 + 1.7524i 0.2417 +1.7872i 1.6464 + 0.7428i 1.3225 + 0.1320i w3 −0.2635 + 1.3614i 0.2274 +1.4208i 0.1966 + 1.4478i 1.3245 + 0.9414i 1.2742 + 0.3922i w4 −0.9918 +1.4880i 0.8678 + 1.2487i 0.9185 + 1.2490i 0.7198 + 1.2427i 1.2901 +1.0495i w5 −0.8109 + 1.2116i 0.7275 + 1.1667i 0.7448 + 1.1524i 0.8106 +1.0040i 1.4625 + 0.7740i w6 −0.8109 + 1.2116i 0.8747 + 1.0470i 0.9536 +0.9516i 0.5595 + 1.0317i 1.0382 + 0.8623i w7 −0.7744 + 1.1516i 0.7930 +1.0406i 0.8912 + 0.9461i 0.6118 + 0.9722i 1.1794 + 0.6376i w8 −0.2340 +0.9857i 0.2098 + 0.9768i 0.2553 + 0.9993i 1.6768 + 0.2002i 0.9430 +0.1100i w9 −0.2349 + 1.0112i 0.2241 + 1.0454i 0.2988 + 1.0689i 0.9997 +0.6844i 0.9069 + 0.2829i w10 −0.2349 + 1.0112i 0.1858 + 0.9878i 0.1656 +1.0288i 1.4212 + 0.4769i 1.0854 + 0.1139i w11 −0.2356 + 1.0372i 0.1901 +1.0659i 0.1779 + 1.1140i 1.1479 + 0.6312i 1.0441 + 0.3296i w12 −0.5320 +0.8680i 0.5547 + 0.8312i 0.5802 + 0.8040i 0.6079 + 0.6566i 0.7273 +0.6160i w13 −0.5496 + 0.8870i 0.5479 + 0.8651i 0.5788 + 0.8534i 0.7284 +0.6957i 0.8177 + 0.4841i w14 −0.5496 + 0.8870i 0.6073 + 0.8182i 0.6616 +0.7612i 0.5724 + 0.7031i 0.8504 + 0.7217i w15 −0.5674 + 0.9058i 0.5955 +0.8420i 0.6574 + 0.7871i 0.6302 + 0.7259i 0.9638 + 0.5407i w16 −1.7549 +0.3495i 1.4070 + 0.1790i 1.4079 + 0.1358i 0.1457 + 1.4010i 0.1658 +1.6747i w17 −1.4293 + 0.2804i 1.7227 + 0.2900i 1.7492 + 0.2856i 0.1866 +1.7346i 0.4907 + 1.6084i w18 −1.4293 + 0.2804i 1.3246 + 0.2562i 1.3108 +0.2733i 0.1174 + 1.1035i 0.1322 + 1.3631i w19 −1.3614 + 0.2635i 1.3636 +0.3654i 1.3393 + 0.4031i 0.1095 + 1.0132i 0.3929 + 1.3102i w20 −1.4880 +0.9918i 1.3708 + 1.2834i 1.3733 + 1.2596i 0.4357 + 1.3636i 1.0646 +1.2876i w21 −1.2116 + 0.8109i 1.6701 + 0.8403i 1.6601 + 0.8198i 0.5853 +1.6820i 0.7949 + 1.4772i w22 −1.2116 + 0.8109i 1.1614 + 0.7909i 1.1559 +0.7249i 0.3439 + 1.0689i 0.8555 + 1.0542i w23 −1.1516 + 0.7744i 1.2241 +0.7367i 1.2163 + 0.6897i 0.3234 + 0.9962i 0.6363 + 1.2064i w24 −0.9857 +0.2340i 0.9769 + 0.1863i 0.9601 + 0.1547i 0.1092 + 0.6174i 0.1088 +0.9530i w25 −1.0112 + 0.2349i 0.9452 + 0.2057i 0.9220 + 0.1683i 0.1074 +0.6307i 0.2464 + 0.9270i w26 −1.0112 + 0.2349i 1.0100 + 0.2182i 1.0004 +0.1894i 0.1109 + 0.6996i 0.1124 + 1.1327i w27 −1.0372 + 0.2356i 0.9795 +0.2417i 0.9581 + 0.2045i 0.1076 + 0.7345i 0.3160 + 1.0913i w28 −0.8680 +0.5320i 0.8241 + 0.4856i 0.8067 + 0.4374i 0.3291 + 0.6264i 0.5707 +0.7662i w29 −0.8870 + 0.5496i 0.8232 + 0.4837i 0.8071 + 0.4310i 0.3126 +0.6373i 0.4490 + 0.8461i w30 −0.8870 + 0.5496i 0.8799 + 0.5391i 0.8603 +0.4941i 0.3392 + 0.6999i 0.6961 + 0.8850i w31 −0.9058 + 0.5674i 0.8796 +0.5356i 0.8598 + 0.4876i 0.3202 + 0.7282i 0.5229 + 1.0037i w32 −0.1570 +0.3085i 0.1376 + 0.3342i 0.1222 + 0.4300i 0.9652 + 0.1066i 0.3237 +0.0849i w33 −0.1564 + 0.3061i 0.1383 + 0.3292i 0.1222 + 0.4272i 0.9075 +0.1666i 0.3228 + 0.0867i w34 −0.1564 + 0.3061i 0.1363 + 0.3322i 0.1213 +0.4297i 0.9724 + 0.1171i 0.4582 + 0.1123i w35 −0.1557 + 0.3037i 0.1370 +0.3273i 0.1213 + 0.4269i 0.9186 + 0.1752i 0.4545 + 0.1251i w36 −0.1726 +0.3073i 0.1655 + 0.3265i 0.1716 + 0.3987i 0.6342 + 0.1372i 0.2844 +0.1296i w37 −0.1716 + 0.3050i 0.1656 + 0.3227i 0.1708 + 0.3972i 0.6550 +0.1495i 0.2853 + 0.1309i w38 −0.1716 + 0.3050i 0.1634 + 0.3246i 0.1707 +0.3976i 0.6290 + 0.1393i 0.3734 + 0.2560i w39 −0.1707 + 0.3028i 0.1636 +0.3208i 0.1700 + 0.3962i 0.6494 + 0.1504i 0.3799 + 0.2517i w40 −0.2066 +0.6354i 0.1779 + 0.6841i 0.1825 + 0.7468i 1.3127 + 0.1240i 0.7502 +0.1138i w41 −0.2071 + 0.6404i 0.1828 + 0.6845i 0.1898 + 0.7378i 0.9572 +0.4344i 0.7325 + 0.2088i w42 −0.2071 + 0.6404i 0.1745 + 0.6828i 0.1588 +0.7521i 1.2403 + 0.2631i 0.6473 + 0.1138i w43 −0.2075 + 0.6460i 0.1793 +0.6829i 0.1666 + 0.7410i 1.0254 + 0.4130i 0.6339 + 0.1702i w44 −0.3219 +0.5854i 0.3547 + 0.6009i 0.3967 + 0.6104i 0.6096 + 0.4214i 0.5902 +0.4857i w45 −0.3250 + 0.5893i 0.3593 + 0.6011i 0.3971 + 0.6093i 0.6773 +0.4284i 0.6355 + 0.4185i w46 −0.3250 + 0.5893i 0.3576 + 0.5990i 0.4022 +0.5964i 0.5995 + 0.4102i 0.4968 + 0.3947i w47 −0.3285 + 0.5937i 0.3624 +0.5994i 0.4020 + 0.5952i 0.6531 + 0.4101i 0.5231 + 0.3644i w48 −0.3085 +0.1570i 0.2697 + 0.1443i 0.2177 + 0.1304i 0.1250 + 0.1153i 0.0872 +0.1390i w49 −0.3061 + 0.1564i 0.2704 + 0.1433i 0.2168 + 0.1304i 0.1252 +0.1158i 0.0871 + 0.1392i w50 −0.3061 + 0.1564i 0.2644 + 0.1442i 0.2184 +0.1305i 0.1245 + 0.1152i 0.0928 + 0.3970i w51 −0.3037 + 0.1557i 0.2650 +0.1432i 0.2174 + 0.1306i 0.1247 + 0.1156i 0.0937 + 0.3973i w52 −0.3073 +0.1726i 0.2763 + 0.1638i 0.2264 + 0.1539i 0.3768 + 0.1244i 0.1053 +0.1494i w53 −0.3050 + 0.1716i 0.2768 + 0.1626i 0.2255 + 0.1538i 0.3707 +0.1237i 0.1052 + 0.1495i w54 −0.3050 + 0.1716i 0.2715 + 0.1630i 0.2269 +0.1542i 0.3779 + 0.1260i 0.1938 + 0.3621i w55 −0.3028 + 0.1707i 0.2719 +0.1618i 0.2260 + 0.1541i 0.3717 + 0.1252i 0.1909 + 0.3627i w56 −0.6354 +0.2066i 0.6488 + 0.1696i 0.6100 + 0.1390i 0.1161 + 0.3693i 0.1091 +0.7656i w57 −0.6404 + 0.2071i 0.6462 + 0.1706i 0.6146 + 0.1379i 0.1157 +0.3645i 0.1699 + 0.7537i w58 −0.6404 + 0.2071i 0.6456 + 0.1745i 0.5973 +0.1418i 0.1176 + 0.3469i 0.1054 + 0.5979i w59 −0.6460 + 0.2075i 0.6431 +0.1753i 0.6021 + 0.1407i 0.1171 + 0.3424i 0.1230 + 0.5949i w60 −0.5854 +0.3219i 0.5854 + 0.3186i 0.5671 + 0.2989i 0.3530 + 0.3899i 0.4294 +0.6363i w61 −0.5893 + 0.3250i 0.5862 + 0.3167i 0.5699 + 0.2952i 0.3422 +0.3808i 0.3744 + 0.6744i w62 −0.5893 + 0.3250i 0.5864 + 0.3275i 0.5622 +0.3050i 0.3614 + 0.3755i 0.3224 + 0.5236i w63 −0.5937 + 0.3285i 0.5873 +0.3254i 0.5648 + 0.3012i 0.3509 + 0.3656i 0.3016 + 0.5347i w/code rate10/15 11/15 12/15 13/15 w0 1.6097 + 0.1548i 0.1512 + 1.5761i 1.1980 +1.1541i 1.1014 + 1.1670i w1 1.5549 + 0.4605i 0.4510 + 1.5251i 0.9192 +1.2082i 0.8557 + 1.2421i w2 1.3226 + 0.1290i 0.1269 + 1.3158i 1.2778 +0.8523i 1.2957 + 0.8039i w3 1.2772 + 0.3829i 0.3773 + 1.2728i 1.0390 +0.9253i 1.0881 + 0.8956i w4 1.2753 + 1.0242i 1.0045 + 1.2537i 0.6057 +1.2200i 0.5795 + 1.2110i w5 1.4434 + 0.7540i 0.7386 + 1.4164i 0.7371 +1.4217i 0.6637 + 1.4215i w6 1.0491 + 0.8476i 0.8392 + 1.0508i 0.6678 +1.0021i 0.6930 + 1.0082i w7 1.1861 + 0.6253i 0.6172 + 1.1844i 0.8412 +0.9448i 0.8849 + 0.9647i w8 0.9326 + 0.0970i 0.0939 + 0.9317i 1.2128 +0.5373i 1.2063 + 0.5115i w9 0.8962 + 0.2804i 0.2781 + 0.8975i 1.0048 +0.5165i 1.0059 + 0.4952i w10 1.1044 + 0.1102i 0.1093 + 1.1073i 1.4321 +0.6343i 1.4171 + 0.5901i w11 1.0648 + 0.3267i 0.3230 + 1.0701i 1.0245 +0.7152i 1.0466 + 0.6935i w12 0.7325 + 0.6071i 0.6074 + 0.7404i 0.6384 +0.6073i 0.6639 + 0.6286i w13 0.8260 + 0.4559i 0.4499 + 0.8324i 0.8175 +0.5684i 0.8353 + 0.5851i w14 0.8744 + 0.7153i 0.7128 + 0.8852i 0.6568 +0.7801i 0.6879 + 0.8022i w15 0.9882 + 0.5300i 0.5259 + 0.9957i 0.8311 +0.7459i 0.8634 + 0.7622i w16 0.1646 + 1.6407i 1.6163 + 0.1632i 0.1349 +1.4742i 0.1213 + 1.4366i w17 0.4867 + 1.5743i 1.5486 + 0.4813i 0.1105 +1.2309i 0.1077 + 1.2098i w18 0.1363 + 1.3579i 1.3596 + 0.1381i 0.0634 +0.9796i 0.0651 + 0.9801i w19 0.4023 + 1.3026i 1.3018 + 0.4076i 0.1891 +1.0198i 0.2009 + 1.0115i w20 1.0542 + 1.2584i 1.2347 + 1.0376i 0.4142 +1.4461i 0.3764 + 1.4264i w21 0.7875 + 1.4450i 1.4193 + 0.7769i 0.3323 +1.2279i 0.3237 + 1.2130i w22 0.8687 + 1.0407i 1.0335 + 0.8734i 0.4998 +0.9827i 0.5205 + 0.9814i w23 0.6502 + 1.1951i 1.1905 + 0.6562i 0.3467 +1.0202i 0.3615 + 1.0163i w24 0.0982 + 0.9745i 0.9948 + 0.1006i 0.0680 +0.6501i 0.0715 + 0.6596i w25 0.2842 + 0.9344i 0.9492 + 0.2955i 0.2016 +0.6464i 0.2116 + 0.6597i w26 0.1142 + 1.1448i 1.1596 + 0.1180i 0.0719 +0.8075i 0.0729 + 0.8131i w27 0.3385 + 1.0973i 1.1091 + 0.3468i 0.2088 +0.8146i 0.2158 + 0.8246i w28 0.6062 + 0.7465i 0.7476 + 0.6205i 0.4809 +0.6296i 0.5036 + 0.6467i w29 0.4607 + 0.8538i 0.8645 + 0.4711i 0.3374 +0.6412i 0.3526 + 0.6572i w30 0.7263 + 0.8764i 0.8777 + 0.7384i 0.4955 +0.8008i 0.5185 + 0.8086i w31 0.5450 + 1.0067i 1.0128 + 0.5570i 0.3431 +0.8141i 0.3593 + 0.8245i w32 0.2655 + 0.0746i 0.0723 + 0.2545i 1.2731 +0.1108i 1.2545 + 0.1010i w33 0.2664 + 0.0759i 0.0761 + 0.2566i 1.0794 +0.0977i 1.0676 + 0.0956i w34 0.4571 + 0.0852i 0.0729 + 0.4512i 1.5126 +0.1256i 1.4782 + 0.1167i w35 0.4516 + 0.1062i 0.1145 + 0.4434i 0.9029 +0.0853i 0.8981 + 0.0882i w36 0.2559 + 0.1790i 0.2114 + 0.2443i 0.5429 +0.0694i 0.5518 + 0.0690i w37 0.2586 + 0.1772i 0.2043 + 0.2508i 0.6795 +0.0559i 0.6903 + 0.0552i w38 0.3592 + 0.2811i 0.3079 + 0.3516i 0.5628 +0.1945i 0.5742 + 0.1987i w39 0.3728 + 0.2654i 0.2673 + 0.3812i 0.7326 +0.1410i 0.7374 + 0.1564i w40 0.7706 + 0.0922i 0.0813 + 0.7729i 1.2283 +0.3217i 1.2378 + 0.3049i w41 0.7407 + 0.2260i 0.2345 + 0.7412i 1.0269 +0.3261i 1.0518 + 0.3032i w42 0.6180 + 0.0927i 0.0756 + 0.6187i 1.4663 +0.3716i 1.4584 + 0.3511i w43 0.6019 + 0.1658i 0.1815 + 0.5946i 0.9085 +0.2470i 0.9107 + 0.2603i w44 0.6007 + 0.4980i 0.5118 + 0.6039i 0.6160 +0.4549i 0.6321 + 0.4729i w45 0.6673 + 0.3928i 0.3821 + 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0.2840 + 0.0610i 0.0654 + 0.0671i 0.0470 + 0.0503i0.0352 + 0.0762i w214 0.2835 + 0.0611i 0.2866 + 0.1555i 0.1636 + 0.3085i0.3374 + 0.1692i w215 0.2840 + 0.0610i 0.2872 + 0.1564i 0.1629 + 0.3082i0.2914 + 0.1413i w216 0.4066 + 0.1836i 0.3982 + 0.3161i 0.3355 + 0.4003i0.4468 + 0.3693i w217 0.4027 + 0.1835i 0.4005 + 0.3136i 0.3227 + 0.4102i0.4545 + 0.3221i w218 0.4066 + 0.1831i 0.3333 + 0.2586i 0.2750 + 0.3318i0.3539 + 0.2625i w219 0.4027 + 0.1830i 0.3330 + 0.2579i 0.2722 + 0.3342i0.3954 + 0.2905i w220 0.2842 + 0.1841i 0.4515 + 0.2301i 0.2346 + 0.4627i0.5230 + 0.2380i w221 0.2847 + 0.1841i 0.4493 + 0.2334i 0.2495 + 0.4548i0.4892 + 0.2652i w222 0.2842 + 0.1835i 0.3681 + 0.1950i 0.2020 + 0.3827i0.3945 + 0.1942i w223 0.2847 + 0.1836i 0.3693 + 0.1954i 0.2054 + 0.3804i0.4344 + 0.2178i w224 0.0567 + 0.4509i 0.8318 + 0.0442i 0.0421 + 0.8420i0.8751 + 0.0419i w225 0.0566 + 0.4510i 0.8239 + 0.1192i 0.1247 + 0.8342i0.8676 + 0.1257i w226 0.0568 + 0.4376i 0.7562 + 0.0454i 0.0384 + 0.7648i0.7986 + 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0.2962 + 0.6193i0.6601 + 0.3001i w255 0.2861 + 0.3130i 0.5877 + 0.3267i 0.3481 + 0.5915i0.6312 + 0.3619i

Next, a definition of the QQAM constellation shall be provided. Eachinput cell word (y₀, . . . , y_(m-1)) shall be modulated using anon-uniform QQAM constellations to give a constellation point zq priorto normalization, where m corresponds to the number of bits per QAMsymbol m=log₂(M). The vector of complex constellation pointsx_(0 . . . M-1) for all combinations of the input bits y_(0 . . . m-1)(corresponding to the decimal values 0 to M−1) are given in the aboveshown tables for the various constellation sizes depending on the QQAMposition vector w_(0 . . . b-1), which defines the constellation pointpositions of a first quarter of the non-uniform constellation. Thelength b of the QQAM position vector w is defined by b=M/4. The QQAMposition vector defines a first quarter of the constellation, namely theconstellation points with the decimal values 0 (y_(0 . . . m)=0000 forthe example of a 16-QQAM) to b−1 (y_(0 . . . m)=0011 for the example ofa 16-QQAM), while the remaining constellation points are derived asfollows:x _(0 . . . b-1)  (first quarter)x _(b . . . 2b-1)=conj(w _(0 . . . b-1))  (second quarter)x _(2b . . . 3b-1)=−conj(w _(0 . . . b-1))  (third quarter)x _(3b . . . 4b-1) =−w _(0 . . . b-1)  (fourth quarter)with conj being the complex conjugate. For example, the correspondingconstellation point z_(q) for a 16-QQAM defined by the QQAM positionvector (w_(0 . . . 3))=(0.2663+0.4530i 0.4530+0.2663i 0.5115+1.2092i1.2092+0.5115i) and the input cell word (y₀ . . . y_(m-1))=(1100) isx₁₂=−w₀=−0.2663−0.4530i. The complete constellation for this NUCposition vector (optimized for code rate 6/15) is shown in the FIG. 7with all input cell words marked at the corresponding constellationpoints.

The QQAM and the N²-NUCconstellations have been defined in such a waythat the bit-wise mutual information is increasing with the bitposition, i.e. the MSB (leftmost bit label) carries the largest mutualinformation, while the LSB (rightmost bit label) carries the leastmutual information. As mentioned above the constellation position vectorw as defined herein does not necessarily contain the constellationpoints of a quadrant, e.g. the first quadrant, of the constellation, butcould also contain the constellation points of any of the four quadrantsor a quarter of constellation points which are not all located in asingle quadrant. Due to the symmetry this leads to constellations with adifferent bit mapping but with identical performance. The constellationposition vector w in the tables defined herein should therefore beconsidered as an example for all four symmetric constellations withdifferent bit mapping but identical performance.

Using N²-QAM constellations it is meaningful from an informationtheoretic point of view to use high constellation orders, since theseconstellations offer more degrees of freedom for the optimization andperform closer to the Shannon capacity as depicted in FIG. 8 . However,with increasing constellation size the complexity for demapping in thereceiver also increases. Since for large N²-QAM constellations manyconstellation points are very close to each other in the complex planeit is proposed in Jonathan Stott, “CM and BICM limits for rectangularconstellations”, DVB document server, document TM-MIMO0007, August 2012,to “condense” non-uniform constellations by means of forcing particularconstellation points to have the same position before the optimizationprocess, accepting a small performance loss compared to its “motherconstellation”. Such constellations are called there “ConQAM” (condensedQAM) and the condensed constellations of QQAM constellations are calledthere “ConQQAM”. This provides a reduced complexity during theoptimization process, since fewer degrees of freedom have to beoptimized and a reduced complexity for demapping in the receiver, due tothe reduced number of “effective” constellation points. In the abovementioned document of Jonathan Stott a condensed 16 kQAM has beenpresented with only 3600 remaining constellation point positions,offering a good performance in the SNR region from 20 to 25 dB.

When the condensation is performed before the optimization, assumptionsmust be made, how a good performing constellation may look like (i.e.which particular points are condensed and which not). This requires adeep analysis for high constellation sizes. Based on these assumptionsof the chosen structure of the constellation, the optimization iscarried out over an SNR region with the corresponding number ofconstellation points (e.g. 268 condensed constellation points instead of1024). The drawback of this approach is that the optimal structure ofthe constellation practically changes for each SNR value, which cannotbe taken into account. That is the resulting ConQAM constellation with afixed number of constellations points is not optimal over a broad SNRrange. Therefore different structures are herein derived and optimized.

An improved alternative to condensing the constellation before theoptimization is the reduction of the constellation points after theoptimization which is proposed according to the present disclosure. Theoptimization of all degrees of freedom of the N²-QAM constellation isthus required, but several advantages are obtained. When performing thecondensation after the optimization, a constellation requiring theminimum required number of constellation points can be derived to offera desired performance. This allows for a seamless change of the requirednumber of constellation points over the SNR range, which leads to areduction of the number of constellation points compared to the approachproposed in the above mentioned document of Jonathan Stott. Thisapproach will be called dynamic condensation, since it is carried outfor each SNR point individually. This approach is outlined for theN²-QAM case in the following.

An example of the algorithm is shown in FIG. 9 for 17 constellationpoints of a PAM constellation: The constellation points with a distancesmaller than the threshold t result in a group of constellation points,i.e. are condensed to a single constellation point position. In the endonly 6 constellation points are remaining. Of course, the algorithm cananalogously be extended to the 2D-case as will be briefly explainedbelow.

The required number of constellation points of the dynamic approach isclearly lower, in addition guaranteeing a maximum performance penaltywith respect to the mother constellation. This leads to a reduced numberof constellation points, further reducing the complexity in thedemapper.

In an embodiment modulator of the disclosed coding and modulationapparatus modulates said cell words into constellation values of anon-uniform constellation wherein said modulator is configured to use,based on the total number M of constellation points of the constellationand the code rate, a non-uniform constellation from a group B ofconstellations comprising one or more of the following constellationsdefined by the constellation position vector u of length v=sqrt(M)/2−1,wherein in one or more constellation position vectors of theconstellations from group B two or more constellation positions areidentical resulting from a condensation of preliminary constellationpositions optimized before.

The following non-uniform constellations comprised in group B areproposed:

B) Condensed M-QAM Non-Uniform Constellations of Group B:

B1) 256-ConQAM NUC u/code rate 6/15 7/15 8/15 9/15 u1 1 1 1 1 u2 2.28382.6712 2.6867 2.6537 u3 2.2838 2.6712 3.0876 3.3114 u4 4.2938 4.67184.8578 4.9478 u5 4.7065 5.3606 5.9639 6.2057 u6 6.5754 7.387 7.95998.1648 u7 8.6463 9.7667 10.4826 10.6773

B2) 1024-ConQAM NUC u/code rate 5/15 6/15 7/15 8/15 9/15 10/15 u1 1       1       1       1       1       1      u2  1       1      1       2.0888  2.5646  2.7751 u3  1       1       1       2.0888 2.5646  2.7751 u4  2.4714  2.7615  2.6688  3.9929  4.55    4.8138 u5 2.4714  2.7615  2.6688  3.9929  4.55    4.8138 u6  2.4714  2.7615 3.2331  5.3856  6.3402  6.878  u7  2.4714  2.7615  3.2331  5.3856 6.3402  6.878  u8  4.4808  4.735   4.9096  7.5592  8.6388  8.8948 u9 4.4808  4.735   4.9096  7.5592  8.6388  9.9366 u10  4.9849  5.5675 6.1134  9.3349 10.5992 11.5652 u11  4.9849  5.5675  6.1134  9.840611.6813 13.0327 u12  6.9079  7.3286  7.7771 11.9228 13.7177 15.0299 u13 6.9079  7.819   8.5339 13.3931 15.6096 17.0928 u14  8.576   9.560810.3128 15.8944 18.2724 19.7629 u15 10.4874 11.6967 12.5661 19.154621.7521 23.229 

B3) 4096-ConQAM NUC u/code rate 5/15 6/15 7/15 8/15 9/15 10/15 11/1512/15 u1  1       1       1       1       1       1       1       1     u2  1       1       1       1       1       2.2716  2.7361  2.8773 u3 1       1       1       1       1       2.2716  2.7361  2.8773 u4 1       1       2.039   2.5335  2.7564  4.1311  4.6854  4.8671 u5 1       1       2.039   2.5335  2.7564  4.1311  4.6854  4.8671 u6 1       1       2.039   2.5335  2.7564  5.4628  6.4578  6.804  u7 1       1       2.039   2.5335  2.7564  5.4628  6.4578  6.804  u8 2.632   2.6741  3.9335  4.5125  4.7808  7.4745  8.5074  8.5047 u9 2.632   2.6741  3.9335  4.5125  4.7808  7.4745  8.5074  9.3098 u10 2.632   2.6741  3.9335  4.5125  4.7808  8.9342 10.3862 10.538  u11 2.632   2.6741  3.9335  4.5125  4.7808  8.9342 10.3862 11.4018 u12 2.632   3.2045  5.2825  6.2741  6.7922 10.9126 12.5386 12.7315 u13 2.632   3.2045  5.2825  6.2741  6.7922 10.9126 12.5386 13.671  u14 2.632   3.2045  5.2825  6.2741  6.7922 12.5824 14.7019 14.9604 u15 2.632   3.2045  5.2825  6.2741  6.7922 12.5824 14.7019 15.9972 u16 4.6312  4.8953  7.4395  8.5533  8.7969 14.7996 17.2181 17.4143 u17 4.6312  4.8953  7.4395  8.5533  8.7969 14.7996 17.2181 18.5456 u18 4.6312  4.8953  7.4395  8.5533  9.7606 16.7338 19.7895 19.9493 u19 4.6312  4.8953  7.4395  8.5533  9.7606 16.7338 19.7895 21.1969 u20 5.2613  6.077   9.1955 10.4959 11.3919 19.1103 22.0532 22.741  u21 5.2613  6.077   9.1955 10.4959 11.3919 19.1103 23.3808 24.129  u22 5.2613  6.077   9.6429 11.4947 12.8073 21.6213 25.1169 25.7517 u23 5.2613  6.077   9.6429 11.4947 12.8073 21.6213 26.6987 27.3256 u24 7.0507  7.7374 11.7079 13.5198 14.7968 24.1315 28.6925 29.1651 u25 7.0507  7.7374 11.7079 13.5198 14.7968 25.5633 30.5564 30.9655 u26 7.4269  8.4608 13.0989 15.1128 16.5067 27.6414 32.7592 33.0129 u27 7.4269  8.4608 13.0989 15.6457 17.5025 29.6431 35.04   35.1483 u28 8.869   9.9898 15.1744 17.6029 19.3416 32.2532 37.7468 37.6166 u29 9.1641 10.497  16.2183 19.0971 21.1053 35.0366 40.6819 40.3063 u3010.5034 12.1004 18.4967 21.5328 23.5673 38.5846 44.2782 43.5432 u3112.3418 14.1335 21.4623 24.7544 26.8076 43.1809 48.8556 47.6401

When optimizing non-uniform QAM constellations, some of theconstellation points tend to merge. This can be exploited bydeliberately merging constellation points that lie close to each otherin order to reduce the complexity in the QAM demapper (but also in theQAM mapper), by simplifying the calculation of the soft-decision loglikelihood ratios (LLRs). Such constellations are called condensed QAMconstellations. If chosen carefully, the loss in performance compared tonon-condensed non-uniform constellations can be neglected. As anexample, the 1024-QQAM constellation optimized for code rate 6/15 can becondensed to 268 constellation point positions reducing the demappingcomplexity while retaining the performance.

In an embodiment the modulator of the disclosed coding and modulationapparatus modulates said cell words into constellation values of anon-uniform constellation wherein said modulator is configured to use,based on the total number M of constellation points of the constellationand the code rate, a non-uniform constellation from a group D ofconstellations comprising one or more of the following constellations,wherein the constellation points are defined by a constellation positionvector w_(0 . . . b-1) with b=M/4, wherein

a first quarter of the constellation points x_(0 . . . b-1) are definedas x_(0 . . . b-1)=w_(0 . . . b-1),

a second quarter of the constellation points x_(b . . . 2b-1) aredefined as x_(b . . . 2b-1)=conj(w_(0 . . . b-1)),

a third quarter of the constellation points x_(2b . . . 3b-1) aredefined as x_(2b . . . 3b-1)=−conj(w_(0 . . . b-1)), and a fourthquarter of the constellation points x_(3b . . . 4b-1) are defined asx_(3b . . . 4b-1)=−w_(0 . . . b-1), wherein conj is the complexconjugate,wherein in one or more constellation position vectors of theconstellations from group D two or more constellation positions areidentical resulting from a condensation of preliminary constellationpositions optimized before.

The following non-uniform constellations comprised in group D areproposed:

D) Condensed M-QAM Non-Uniform Constellations of Group D:

D1) 64-ConQQAM NUC w/code rate 5/15 6/15 7/15 8/15 9/15 w0 1.0257 +0.5960i 0.5656 + 0.9499i 0.2925 + 1.4892i 0.2920 + 1.4827i 0.2978 +1.4669i w1 1.2181 + 0.7476i 0.2750 + 1.0676i 0.8449 + 1.2622i 0.8411 +1.2563i 0.8421 + 1.2355i w2 1.1509 + 0.3069i 0.8202 + 1.2651i 0.2351 +1.0196i 0.2174 + 1.0211i 0.2135 + 1.0389i w3 1.3888 + 0.3325i 0.3011 +1.4529i 0.5555 + 0.8926i 0.5702 + 0.8798i 0.6055 + 0.8654i w4 0.5961 +1.0257i 0.9500 + 0.5641i 1.4892 + 0.2925i 1.4827 + 0.2920i 1.4685 +0.2859i w5 0.7476 + 1.2181i 1.0666 + 0.2744i 1.2622 + 0.8449i 1.2563 +0.8410i 1.2516 + 0.8201i w6 0.3069 + 1.1510i 1.2657 + 0.8178i 1.0196 +0.2351i 1.0211 + 0.2174i 1.0279 + 0.1981i w7 0.3325 + 1.3888i 1.4521 +0.3005i 0.8926 + 0.5555i 0.8798 + 0.5702i 0.8857 + 0.5642i w8 0.4969 +0.2685i 0.2717 + 0.5512i 0.1635 + 0.3025i 0.1583 + 0.3034i 0.1545 +0.3665i w9 0.4123 + 0.2376i 0.2717 + 0.5512i 0.1635 + 0.3025i 0.1583 +0.3034i 0.1545 + 0.3665i w10 0.4969 + 0.2685i 0.2020 + 0.3572i 0.2075 +0.6586i 0.1871 + 0.6855i 0.1756 + 0.7261i w11 0.4123 + 0.2376i 0.2020 +0.3572i 0.3354 + 0.6030i 0.3563 + 0.6126i 0.4023 + 0.6180i w12 0.2685 +0.4969i 0.5503 + 0.2716i 0.3025 + 0.1635i 0.3034 + 0.1583i 0.2731 +0.1455i w13 0.2376 + 0.4123i 0.5503 + 0.2716i 0.3025 + 0.1635i 0.3034 +0.1583i 0.2731 + 0.1455i w14 0.2685 + 0.4969i 0.3561 + 0.2021i 0.6586 +0.2075i 0.6855 + 0.1871i 0.6840 + 0.1578i w15 0.2376 + 0.4123i 0.3561 +0.2021i 0.6030 + 0.3354i 0.6126 + 0.3563i 0.6145 + 0.3556i

D2) 256-ConQQAM NUC w/code 5/15 6/15 7/15 8/15 9/15 10/15 11/15 w0−0.3495 + 1.7549i 0.6800 + 1.6926i 0.7280 + 1.6384i 1.0804 + 1.3788i1.6350 + 0.1593i 1.6097 + 0.1548i 0.1512 + 1.5761i w1 −0.2804 + 1.4293i0.3911 + 1.3645i 0.4787 + 1.3492i 1.0487 + 0.9862i 1.5776 + 0.4735i1.5549 + 0.4605i 0.4510 + 1.5251i w2 −0.2804 + 1.4293i 0.2191 + 1.7524i0.2417 + 1.7872i 1.6464 + 0.7428i 1.3225 + 0.1320i 1.3226 + 0.1290i0.1269 + 1.3158i w3 −0.2635 + 1.3614i 0.2274 + 1.4208i 0.1966 + 1.4478i1.3245 + 0.9414i 1.2742 + 0.3922i 1.2772 + 0.3829i 0.3773 + 1.2728i w4−0.9918 + 1.4880i 0.8678 + 1.2487i 0.9185 + 1.2490i 0.7198 + 1.2427i1.2901 + 1.0495i 1.2753 + 1.0242i 1.0045 + 1.2537i w5 −0.8109 + 1.2116i0.7275 + 1.1667i 0.7448 + 1.1524i 0.8106 + 1.0040i 1.4625 + 0.7740i1.4434 + 0.7540i 0.7386 + 1.4164i w6 −0.8109 + 1.2116i 0.8747 + 1.0470i0.9536 + 0.9516i 0.5595 + 1.0317i 1.0382 + 0.8623i 1.0491 + 0.8476i0.8392 + 1.0508i w7 −0.7744 + 1.1516i 0.7930 + 1.0406i 0.8912 + 0.9461i0.6118 + 0.9722i 1.1794 + 0.6376i 1.1861 + 0.6253i 0.6172 + 1.1844i w8−0.2349 + 1.0113i 0.1978 + 0.9823i 0.2553 + 0.9993i 1.6768 + 0.2002i0.9430 + 0.1100i 0.9326 + 0.0970i 0.0939 + 0.9317i w9 −0.2349 + 1.0113i0.2071 + 1.0557i 0.2988 + 1.0689i 0.9997 + 0.6844i 0.9069 + 0.2829i0.8962 + 0.2804i 0.2781 + 0.8975i w10 −0.2349 + 1.0113i 0.1978 + 0.9823i0.1656 + 1.0288i 1.4212 + 0.4769i 1.0854 + 0.1139i 1.1044 + 0.1102i0.1093 + 1.1073i w11 −0.2349 + 1.0113i 0.2071 + 1.0557i 0.1779 + 1.1140i1.1479 + 0.6312i 1.0441 + 0.3296i 1.0648 + 0.3267i 0.3230 + 1.0701i w12−0.5497 + 0.8869i 0.5660 + 0.8461i 0.5795 + 0.8287i 0.6079 + 0.6566i0.7273 + 0.6160i 0.7325 + 0.6071i 0.6074 + 0.7404i w13 −0.5497 + 0.8869i0.5660 + 0.8461i 0.5795 + 0.8287i 0.7284 + 0.6957i 0.8177 + 0.4841i0.8260 + 0.4559i 0.4499 + 0.8324i w14 −0.5497 + 0.8869i 0.6073 + 0.8182i0.6595 + 0.7742i 0.5724 + 0.7031i 0.8504 + 0.7217i 0.8744 + 0.7153i0.7128 + 0.8852i w15 −0.5497 + 0.8869i 0.5660 + 0.8461i 0.6595 + 0.7742i0.6302 + 0.7259i 0.9638 + 0.5407i 0.9882 + 0.5300i 0.5259 + 0.9957i w16−1.7549 + 0.3495i 1.4070 + 0.1790i 1.4079 + 0.1358i 0.1457 + 1.4010i0.1658 + 1.6747i 0.1646 + 1.6407i 1.6163 + 0.1632i w17 −1.4293 + 0.2804i1.7227 + 0.2900i 1.7492 + 0.2856i 0.1866 + 1.7346i 0.4907 + 1.6084i0.4867 + 1.5743i 1.5486 + 0.4813i w18 −1.4293 + 0.2804i 1.3246 + 0.2562i1.3108 + 0.2733i 0.1174 + 1.1035i 0.1322 + 1.3631i 0.1363 + 1.3579i1.3596 + 0.1381i w19 −1.3614 + 0.2635i 1.3636 + 0.3654i 1.3393 + 0.4031i0.1095 + 1.0132i 0.3929 + 1.3102i 0.4023 + 1.3026i 1.3018 + 0.4076i w20−1.4880 + 0.9918i 1.3708 + 1.2834i 1.3733 + 1.2596i 0.4357 + 1.3636i1.0646 + 1.2876i 1.0542 + 1.2584i 1.2347 + 1.0376i w21 −1.2116 + 0.8109i1.6701 + 0.8403i 1.6601 + 0.8198i 0.5853 + 1.6820i 0.7949 + 1.4772i0.7875 + 1.4450i 1.4193 + 0.7769i w22 −1.2116 + 0.8109i 1.1614 + 0.7909i1.1559 + 0.7249i 0.3439 + 1.0689i 0.8555 + 1.0542i 0.8687 + 1.0407i1.0335 + 0.8734i w23 −1.1516 + 0.7744i 1.2241 + 0.7367i 1.2163 + 0.6897i0.3234 + 0.9962i 0.6363 + 1.2064i 0.6502 + 1.1951i 1.1905 + 0.6562i w24−1.0113 + 0.2349i 0.9774 + 0.2034i 0.9601 + 0.1792i 0.1083 + 0.6241i0.1088 + 0.9530i 0.0982 + 0.9745i 0.9948 + 0.1006i w25 −1.0113 + 0.2349i0.9774 + 0.2034i 0.9601 + 0.1792i 0.1083 + 0.6241i 0.2464 + 0.9270i0.2842 + 0.9344i 0.9492 + 0.2955i w26 −1.0113 + 0.2349i 0.9774 + 0.2034i0.9601 + 0.1792i 0.1093 + 0.7170i 0.1124 + 1.1327i 0.1142 + 1.1448i1.1596 + 0.1180i w27 −1.0113 + 0.2349i 0.9795 + 0.2417i 0.9601 + 0.1792i0.1093 + 0.7170i 0.3160 + 1.0913i 0.3385 + 1.0973i 1.1091 + 0.3468i w28−0.8869 + 0.5497i 0.8236 + 0.4847i 0.8069 + 0.4342i 0.3208 + 0.6318i0.5707 + 0.7662i 0.6062 + 0.7465i 0.7476 + 0.6205i w29 −0.8869 + 0.5497i0.8236 + 0.4847i 0.8069 + 0.4342i 0.3208 + 0.6318i 0.4490 + 0.8461i0.4607 + 0.8538i 0.8645 + 0.4711i w30 −0.8869 + 0.5497i 0.8798 + 0.5374i0.8601 + 0.4908i 0.3297 + 0.7141i 0.6961 + 0.8850i 0.7263 + 0.8764i0.8777 + 0.7384i w31 −0.8869 + 0.5497i 0.8798 + 0.5374i 0.8601 + 0.4908i0.3297 + 0.7141i 0.5229 + 1.0037i 0.5450 + 1.0067i 1.0128 + 0.5570i w32−0.1640 + 0.3056i 0.1509 + 0.3272i 0.1217 + 0.4285i 0.9688 + 0.1119i0.3232 + 0.0858i 0.2660 + 0.0752i 0.0742 + 0.2556i w33 −0.1640 + 0.3056i0.1509 + 0.3272i 0.1217 + 0.4285i 0.9131 + 0.1709i 0.3232 + 0.0858i0.2660 + 0.0752i 0.0742 + 0.2556i w34 −0.1640 + 0.3056i 0.1509 + 0.3272i0.1217 + 0.4285i 0.9688 + 0.1119i 0.4564 + 0.1187i 0.4544 + 0.0957i0.0937 + 0.4473i w35 −0.1640 + 0.3056i 0.1509 + 0.3272i 0.1217 + 0.4285i0.9131 + 0.1709i 0.4564 + 0.1187i 0.4544 + 0.0957i 0.0937 + 0.4473i w36−0.1640 + 0.3056i 0.1509 + 0.3272i 0.1708 + 0.3974i 0.6419 + 0.1441i0.2849 + 0.1302i 0.2572 + 0.1781i 0.2078 + 0.2475i w37 −0.1640 + 0.3056i0.1509 + 0.3272i 0.1708 + 0.3974i 0.6419 + 0.1441i 0.2849 + 0.1302i0.2572 + 0.1781i 0.2078 + 0.2475i w38 −0.1640 + 0.3056i 0.1509 + 0.3272i0.1708 + 0.3974i 0.6419 + 0.1441i 0.3766 + 0.2539i 0.3660 + 0.2732i0.2876 + 0.3664i w39 −0.1640 + 0.3056i 0.1509 + 0.3272i 0.1708 + 0.3974i0.6419 + 0.1441i 0.3766 + 0.2539i 0.3660 + 0.2732i 0.2876 + 0.3664i w40−0.2071 + 0.6406i 0.1786 + 0.6836i 0.1744 + 0.7444i 1.3127 + 0.1240i0.7502 + 0.1138i 0.7706 + 0.0922i 0.0813 + 0.7729i w41 −0.2071 + 0.6406i0.1786 + 0.6836i 0.1744 + 0.7444i 0.9572 + 0.4344i 0.7325 + 0.2088i0.7407 + 0.2260i 0.2345 + 0.7412i w42 −0.2071 + 0.6406i 0.1786 + 0.6836i0.1744 + 0.7444i 1.2403 + 0.2631i 0.6473 + 0.1138i 0.6180 + 0.0927i0.0756 + 0.6187i w43 −0.2071 + 0.6406i 0.1786 + 0.6836i 0.1744 + 0.7444i1.0254 + 0.4130i 0.6339 + 0.1702i 0.6019 + 0.1658i 0.1815 + 0.5946i w44−0.3251 + 0.5895i 0.3585 + 0.6001i 0.3995 + 0.6028i 0.6207 + 0.4139i0.5902 + 0.4857i 0.6007 + 0.4980i 0.5118 + 0.6039i w45 −0.3251 + 0.5895i0.3585 + 0.6001i 0.3995 + 0.6028i 0.6773 + 0.4284i 0.6355 + 0.4185i0.6673 + 0.3928i 0.3821 + 0.6808i w46 −0.3251 + 0.5895i 0.3585 + 0.6001i0.3995 + 0.6028i 0.6207 + 0.4139i 0.5099 + 0.3795i 0.4786 + 0.3935i0.4145 + 0.4734i w47 −0.3251 + 0.5895i 0.3585 + 0.6001i 0.3995 + 0.6028i0.6207 + 0.4139i 0.5099 + 0.3795i 0.5176 + 0.3391i 0.3230 + 0.5340i w48−0.3056 + 0.1640i 0.2707 + 0.1533i 0.2219 + 0.1422i 0.1249 + 0.1155i0.0962 + 0.1443i 0.0755 + 0.1004i 0.0859 + 0.0728i w49 −0.3056 + 0.1640i0.2707 + 0.1533i 0.2219 + 0.1422i 0.1249 + 0.1155i 0.0962 + 0.1443i0.0755 + 0.1004i 0.0859 + 0.0728i w50 −0.3056 + 0.1640i 0.2707 + 0.1533i0.2219 + 0.1422i 0.1249 + 0.1155i 0.0932 + 0.3972i 0.0822 + 0.4771i0.5165 + 0.0795i w51 −0.3056 + 0.1640i 0.2707 + 0.1533i 0.2219 + 0.1422i0.1249 + 0.1155i 0.0932 + 0.3972i 0.0822 + 0.4771i 0.5165 + 0.0795i w52−0.3056 + 0.1640i 0.2707 + 0.1533i 0.2219 + 0.1422i 0.3743 + 0.1248i0.0962 + 0.1443i 0.1016 + 0.2242i 0.2619 + 0.0919i w53 −0.3056 + 0.1640i0.2707 + 0.1533i 0.2219 + 0.1422i 0.3743 + 0.1248i 0.0962 + 0.1443i0.1016 + 0.2242i 0.2619 + 0.0919i w54 −0.3056 + 0.1640i 0.2707 + 0.1533i0.2219 + 0.1422i 0.3743 + 0.1248i 0.1923 + 0.3624i 0.1916 + 0.3944i0.4223 + 0.2036i w55 −0.3056 + 0.1640i 0.2707 + 0.1533i 0.2219 + 0.1422i0.3743 + 0.1248i 0.1923 + 0.3624i 0.1916 + 0.3944i 0.4223 + 0.2036i w56−0.6406 + 0.2071i 0.6459 + 0.1725i 0.6060 + 0.1399i 0.1166 + 0.3558i0.1091 + 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D3) 1024−ConQQAM NUC w/ code rate 5/15 6/15 7/15 8/15 9/15 w0 1.9734 +0.6475i 1.3555 + 1.1873i 0.1119 + 1.7802i 0.0966 + 1.7257i 0.1003 +1.7258i w1 1.7794 + 0.9784i 1.0871 + 1.4324i 0.2324 + 2.0028i 0.1799 +1.9540i 0.1722 + 1.9393i w2 0.4642 + 1.8997i 1.5003 + 1.4954i 0.1416 +1.5629i 0.1216 + 1.5846i 0.0956 + 1.5505i w3 0.8851 + 1.9829i 1.1127 +1.7206i 0.1594 + 1.5242i 0.2925 + 1.5646i 0.2621 + 1.5384i w4 0.2393 +1.3953i 0.2080 + 2.0408i 0.8323 + 1.8356i 0.7016 + 1.7899i 0.6393 +1.7656i w5 0.2393 + 1.3953i 0.7958 + 1.4948i 0.5102 + 1.8340i 0.4407 +1.8294i 0.3969 + 1.7846i w6 0.2534 + 1.4887i 0.5864 + 1.9337i 0.4040 +1.5212i 0.6347 + 1.5255i 0.6050 + 1.5226i w7 0.2393 + 1.3953i 0.7534 +1.6241i 0.4040 + 1.5212i 0.4512 + 1.5701i 0.4221 + 1.5525i w8 2.0528 +0.2197i 1.0007 + 1.0906i 0.1151 + 1.2407i 0.0862 + 1.3505i 0.0799 +1.3199i w9 1.6003 + 1.3185i 0.9660 + 1.1761i 0.1151 + 1.2407i 0.2378 +1.3237i 0.2425 + 1.3020i w10 0.1805 + 2.0047i 0.9193 + 1.0471i 0.1219 +1.3013i 0.0908 + 1.3996i 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0.8676 + 0.1257i w226 0.0567 + 0.4443i 0.7562 + 0.0454i 0.0384 +0.7648i 0.7986 + 0.0381i w227 0.0567 + 0.4443i 0.7500 + 0.1031i 0.1132 +0.7575i 0.7927 + 0.1144i w228 0.1692 + 0.4451i 0.7842 + 0.2789i 0.2858 +0.7945i 0.8293 + 0.2896i w229 0.1692 + 0.4451i 0.8063 + 0.2068i 0.2067 +0.8184i 0.8523 + 0.2091i w230 0.1692 + 0.4451i 0.7167 + 0.2465i 0.2585 +0.7216i 0.7578 + 0.2626i w231 0.1692 + 0.4451i 0.7319 + 0.1921i 0.1886 +0.7429i 0.7786 + 0.1893i w232 0.0568 + 0.3109i 0.5929 + 0.0575i 0.0601 +0.6048i 0.6439 + 0.0332i w233 0.0568 + 0.3109i 0.5929 + 0.0575i 0.0601 +0.6048i 0.6363 + 0.0929i w234 0.0568 + 0.3109i 0.6701 + 0.0660i 0.0378 +0.6868i 0.7212 + 0.0348i w235 0.0568 + 0.3109i 0.6701 + 0.0660i 0.0991 +0.6806i 0.7163 + 0.1032i w236 0.1690 + 0.3113i 0.5701 + 0.1700i 0.1750 +0.5791i 0.6122 + 0.2069i w237 0.1690 + 0.3113i 0.5701 + 0.1700i 0.1750 +0.5791i 0.6268 + 0.1493i w238 0.1690 + 0.3113i 0.6451 + 0.1940i 0.2285 +0.6480i 0.6853 + 0.2349i w239 0.1690 + 0.3113i 0.6451 + 0.1940i 0.1719 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0.2826 +0.5338i 0.5635 + 0.3210i w254 0.2858 + 0.3115i 0.5946 + 0.3133i 0.2962 +0.6193i 0.6601 + 0.3001i w255 0.2858 + 0.3115i 0.5946 + 0.3133i 0.3481 +0.5915i 0.6312 + 0.3619i

For further illustration FIGS. 11A-11C show two of the proposed1024-N²-NUCs optimized for 27 dB SNR (FIG. 11A) and 8 dB SNR (FIG. 11B)and an enlargement of the N²-NUC optimized for 8 dB SNR (FIG. 11C),illustrating that several constellation points tend to merge at low SNR.

FIGS. 12A and 12B show two 64-QQAMs optimized for 10 dB SNR (FIG. 12A)and 15 dB SNR (FIG. 12B). In contrast to the rectangular shaped N²-NUCs,the QQAM constellations have a circular shape, reducing the peak-powerof the constellation compared to N²-NUCs, by avoiding the constellationpoints near the corner. The constellation points of large QQAMs tend tomerge in the same way like N²-NUCs when being optimized at low SNR. Thisis e.g. the case in the inner circle of the 64-QQAM optimized at 10 dBtarget SNR shown in FIG. 12A.

The condensation of constellations, especially if optimized for very lowSNR, may sometimes lead to (complete) puncturing of the leastsignificant bits of the constellation. This is for example the case fora 256-N²-NUC optimized for 0 dB SNR, which results in a QPSKconstellation with all 64 constellation points of one quadrant havingexactly the same constellation point position. When demapping such aconstellation in the receiver, the first two bits can be restored bymeans of the four different constellation point positions. In case of anunencoded system the remaining 6 least significant bits could not berestored. This is however possible using a BICM chain withstate-of-the-art forward error correction codes, which is able tocorrect the remaining bits based on the information of the two mostsignificant bits. Such a BICM chain, as e.g. conventionally used insystems according to various DVB standards, is thus preferably used inthe transmitter and the receiver of a communication system according tothe present disclosure. Preferably, the use of such a BICM chain isassumed to be used when performing an optimization of theconstellations, and the BICM capacity is the target capacity during theoptimization process. In the extreme example mentioned above, it wouldalternatively possible to directly transmit a QPSK constellationcarrying only 2 bits per constellation symbol to avoid the increaseddemapping complexity. For higher SNR the transmission of condensednon-uniform constellations with very high order is however advantageousfrom a performance perspective compared to smaller non-uniformconstellations.

In still another embodiment the modulator of the disclosed coding andmodulation apparatus modulates said cell words into constellation valuesof a non-uniform constellation wherein said modulator is configured touse, based on the total number M of constellation points of theconstellation and the code rate, a non-uniform constellation obtainedfrom a constellation from anyone of groups A, B, C or D through rotationby an angle around the origin. In other words, one or more of thefollowing “invariant transformations” do not affect the properties of amapping:

1. rotation of all symbols by arbitrary angle φ;

2. inversion of m-th bit y_m=b∈{0,1} to y_m=⁻b, where the bar indicatesinversion;

3. interchanging of bit positions y_k1 and y_k2;

4. reflection on Re{x1}- and/or Im{x1}-axis.

In still another embodiment the modulator of the disclosed coding andmodulation apparatus modulates said cell words into constellation valuesof a non-uniform constellation wherein said modulator is configured touse, based on the total number M of constellation points of theconstellation and the code rate, a non-uniform constellation from agroup E of constellations comprising one or more of the followingconstellations defined by the constellation position vectorw_(0 . . . M−). Such constellations can not be described by the symmetryfrom the QQAM in a straightforward manner. Therefore, the completeconstellation position vector with M entries will be used.

The following non-uniform constellations comprised in group E areproposed:

E) M-QAM Non-Uniform Constellations of Group E:

E1) 8-QAM 2D NUC w/code rate 100/180 or 104/180 23/36 25/36 or 13/1832/45 (16200 length code) 19/30 w0 −0.2330 −0.2231 −0.2416 −0.2416 +0.0000i   0.8753 + 0.5486i w1 −0.5414 − 0.8712i −0.5520 − 0.8882i−0.5547 − 0.8925i −0.5547 − 0.8925i +1.3470i w2 −0.5414 + 0.8712i−0.5520 + 0.8882i −0.5547 + 0.8925i −0.5547 + 0.8925i +0.2266i w3−1.3570 −1.3279 −1.3165 −1.3165 + 0.0000i −0.8753 + 0.5486i w4  0.2330 0.2231  0.2416 0.2416   0.8753 − 0.5486i w5 0.5414 − 0.8712i 0.5520 −0.8882i 0.5547 − 0.8925i 0.5547 − 0.8925i −1.3470i w6 0.5414 + 0.8712i0.5520 + 0.8882i 0.5547 + 0.8925i 0.5547 + 0.8925i −0.2266i w7  1.3570 1.3279  1.3165 1.3165 −0.8753 − 0.5486i

E2) 16-QAM 2D NUC w/ 90/180 or 96/180 or code 100/180 26/45 3/5 28/45 or23/36 25/36 or 13/18 140/180 or 154/180 20/30 w0  0.6186 + 0.2544i 0.4718 + 0.2606i  0.4718 + 0.2606i  0.4936 + 0.2530i  0.6186 + 0.2544i 0.6578 + 0.2571i  0.5061 + 0.2474i w1 −0.6186 + 0.2544i −0.4718 +0.2606i −0.4718 + 0.2606i −0.4936 + 0.2530i −0.6186 + 0.2544i −0.6578 +0.2571i  0.2474 + 0.5061i w2  0.6186 − 0.2544i  0.4718 − 0.2606i  0.4718− 0.2606i  0.4936 − 0.2530i  0.6186 − 0.2544i  0.6578 − 0.2571i−0.5061 + 0.2474i w3 −0.6186 − 0.2544i −0.4718 − 0.2606i −0.4718 −0.2606i −0.4936 − 0.2530i −0.6186 − 0.2544i −0.6578 − 0.2571i −0.2474 +0.5061i w4  1.2080 + 0.5377i  1.2088 + 0.4984i  0.2606 + 0.4718i 0.2530 + 0.4936i  1.2080 + 0.5377i  1.2088 + 0.5659i  0.5061 − 0.2474iw5 −1.2080 + 0.5377i −1.2088 + 0.4984i −0.2606 + 0.4718i −0.2530 +0.4936i −1.2080 + 0.5377i −1.2088 + 0.5659i  0.2474 − 0.5061i w6  1.2080− 0.5377i  1.2088 − 0.4984i  0.2606 − 0.4718i  0.2530 − 0.4936i  1.2080− 0.5377i  1.2088 − 0.5659i −0.5061 − 0.2474i w7 −1.2080 − 0.5377i−1.2088 − 0.4984i −0.2606 − 0.4718i −0.2530 − 0.4936i −1.2080 − 0.5377i−1.2088 − 0.5659i −0.2474 − 0.5061i w8  0.2213 + 0.4416i  0.2606 +0.4718i  1.2088 + 0.4984i  1.2040 + 0.4925i  0.2213 + 0.4416i  0.2173 +0.4189i  1.2007 + 0.4909i w9 −0.2213 + 0.4416i −0.2606 + 0.4718i−1.2088 + 0.4984i −1.2040 + 0.4925i −0.2213 + 0.4416i −0.2173 + 0.4189i 0.4909 + 1.2007i w10  0.2213 − 0.4416i  0.2606 − 0.4718i  1.2088 −0.4984i  1.2040 − 0.4925i  0.2213 − 0.4416i  0.2173 − 0.4189i −1.2007 +0.4909i w11 −0.2213 − 0.4416i −0.2606 − 0.4718i −1.2088 − 0.4984i−1.2040 − 0.4925i −0.2213 − 0.4416i −0.2173 − 0.4189i −0.4909 + 1.2007iw12  0.4487 + 1.1657i  0.4984 + 1.2088i  0.4984 + 1.2088i  0.4925 +1.2040i  0.4487 + 1.1657i  0.4326 + 1.1445i  1.2007 − 0.4909i w13−0.4487 + 1.1657i −0.4984 + 1.2088i −0.4984 + 1.2088i −0.4925 + 1.2040i−0.4487 + 1.1657i −0.4326 + 1.1445i  0.4909 − 1.2007i w14  0.4487 −1.1657i  0.4984 − 1.2088i  0.4984 − 1.2088i  0.4925 − 1.2040i  0.4487 −1.1657i  0.4326 − 1.1445i −1.2007 − 0.4909i w15 −0.4487 − 1.1657i−0.4984 − 1.2088i −0.4984 − 1.2088i −0.4925 − 1.2040i −0.4487 − 1.1657i−0.4326 − 1.1445i −0.4909 − 1.2007i

E3) 32-QAM 2D NUC w/code rate 2/3 or 25/36 128/180 or 132/180 or 140/180w0   0.1929 + 0.1744i   0.2121 + 0.1569i w1   0.2283 + 0.5036i −0.2121 +0.1569i w2   0.1929 − 0.1744i  0.2121 − 0.1569i w3   0.2283 − 0.5036i−0.2121 − 0.1569i w4 −0.1929 + 0.1744i   0.7185 + 0.1739i w5 −0.2283 +0.5036i −0.7185 + 0.1739i w6 −0.1929 − 0.1744i   0.7185 − 0.1739i w7−0.2283 − 0.5036i −0.7185 − 0.1739i w8   0.3541 + 1.4168i   0.3478 +1.4027i w9   0.2627 + 0.9170i −0.3478 + 1.4027i w10   0.3541 − 1.4168i  0.3478 − 1.4027i w11   0.2627 − 0.9170i −0.3478 − 1.4027i w12−0.3541 + 1.4168i   1.2867 + 0.3209i w13 −0.2627 + 0.9170i −1.2867 +0.3209i w14 −0.3541 − 1.4168i   1.2867 − 0.3209i w15 −0.2627 − 0.9170i−1.2867 − 0.3209i w16   0.6963 + 0.1782i   0.2258 + 0.5089i w17  0.6364 + 0.4437i −0.2258 + 0.5089i w18   0.6963 − 0.1782i   0.2258 −0.5089i w19   0.6364 − 0.4437i −0.2258 − 0.5089i w20 −0.6963 + 0.1782i  0.6431 + 0.5018i w21 −0.6364 + 0.4437i −0.6431 − 0.5018i w22 −0.6963 −0.1782i   0.6431 − 0.5018i w23 −0.6364 − 0.4437i −0.6431 − 0.5018i w24  1.3162 + 0.3270i   0.2443 + 0.9172i w25   0.9382 + 0.8637i −0.2443 +0.9172i w26   1.3162 − 0.3270i   0.2443 − 0.9172i w27   0.9382 − 0.8637i−0.2443 − 0.9172i w28 −1.3162 + 0.3270i   0.9274 + 0.8949i w29 −0.9382 +0.8637i −0.9274 + 0.8949i w30 −1.3162 − 0.3270i   0.9274 − 0.8949i w31−0.9382 − 0.8637i −0.9274 − 0.8949i

E4) 64-QAM 2D NUC w/code rate 132/180 140/180 or 144/180 7/9 or 4/5150/180 5/6 21/30 23/30 w0   0.2878 + 1.4388i   0.9856 − 0.4661i  0.2775 + 1.4188i   1.0161 − 0.4912i   1.0161 − 0.4912i   0.2994 +1.4627i   0.2878 + 1.4388i w1 −0.2878 + 1.4388i −0.9856 − 0.4661i  0.1138 + 0.3999i   1.0161 + 0.4912i   0.1287 − 0.4061i   0.8421 +1.2279i   0.8133 + 1.2150i w2   0.2878 − 1.4388i   0.9856 + 0.4661i  0.2177 + 1.0243i −1.0161 − 0.4912i   1.0441 − 0.1581i   1.4625 +0.2831i   0.2878 − 1.4388i w3 −0.2878 − 1.4388i −0.9856 + 0.4661i  0.1487 + 0.7260i −1.0161 + 0.4912i   0.1321 − 0.1317i   1.2488 +0.8125i   0.8133 − 1.2150i w4   0.1177 + 0.4119i   1.1580 − 0.8178i  0.7921 + 1.2096i   0.1287 − 0.4061i   0.6966 − 0.4427i   0.1299 +0.4106i −0.2878 + 1.4388i w5 −0.1177 + 0.4119i −1.1580 − 0.8178i  0.2891 + 0.3910i   0.1287 + 0.4061i   0.4025 − 0.4142i   0.1823 +0.3822i −0.8133 + 1.2150i w6   0.1177 − 0.4119i   1.1580 + 0.8178i  0.6056 + 0.8481i −0.1287 − 0.4061i   0.6995 − 0.1411i   0.2512 +0.1287i −0.2878 − 1.4388i w7 −0.1177 − 0.4119i −1.1580 + 0.8178i  0.4397 + 0.5853i −0.1287 + 0.4061i   0.4035 − 0.1354i   0.2598 +0.1538i −0.8133 − 1.2150i w8  1.4656 + 0.2931i  0.1264 − 0.4145i  0.2775− 1.4188i   1.1306 − 0.8649i   1.0161 + 0.4912i −0.2994 + 1.4627i  0.2219 + 1.0386i w9 −1.4656 + 0.2931i −0.1264 − 0.4145i   0.1138 −0.3999i   1.1306 + 0.8649i   0.1287 + 0.4061i −0.8421 + 1.2279i  0.6145 + 0.8494i w10   1.4656 − 0.2931i   0.1264 + 0.4145i   0.2177 −1.0243i −1.1306 − 0.8649i   1.0441 + 0.1581i −1.4625 + 0.2831i   0.2219− 1.0386i w11 −1.4656 − 0.2931i −0.1264 + 0.4145i   0.1487 − 0.7260i−1.1306 + 0.8649i   0.1321 + 0.1317i −1.2488 + 0.8125i   0.6145 −0.8494i w12   0.1678 + 0.1166i   0.1416 − 0.7330i   0.7921 − 1.2096i  0.1385 − 0.7199i   0.6966 + 0.4427i −0.1299 + 0.4106i −0.2219 +1.0386i w13 −0.1678 + 0.1166i −0.1416 − 0.7330i   0.2891 − 0.3910i  0.1385 + 0.7199i   0.4025 + 0.4142i −0.1823 + 0.3822i −0.6145 +0.8494i w14   0.1678 − 0.1166i   0.1416 + 0.7330i   0.6056 − 0.8481i−0.1385 − 0.7199i   0.6995 + 0.1411i −0.2512 + 0.1287i −0.2219 − 1.0386iw15 −0.1678 − 0.1166i −0.1416 + 0.7330i   0.4397 − 0.5853i −0.1385 +0.7199i   0.4035 + 0.1354i −0.2598 + 0.1538i −0.6145 − 0.8494i w16  0.2219 + 1.0386i   1.0366 − 0.1534i   1.4730 + 0.3019i   1.0441 −0.1581i   1.1306 − 0.8649i   0.2994 − 1.4627i   0.1177 + 0.4119i w17−0.2219 + 1.0386i −1.0366 − 0.1534i   0.1419 + 0.1122i   1.0441 +0.1581i   0.1385 − 0.7199i   0.8421 − 1.2279i   0.2516 + 0.3998i w18  0.2219 − 1.0386i   1.0366 + 0.1534i   1.0895 + 0.2172i −1.0441 −0.1581i   1.4516 − 0.2578i   1.4625 − 0.2831i   0.1177 − 0.4119i w19−0.2219 − 1.0386i −1.0366 + 0.1534i   0.7863 + 0.1337i −1.0441 + 0.1581i  0.1689 − 1.0567i   1.2488 − 0.8125i   0.2516 − 0.3998i w20   0.1559 +0.7442i   1.4529 − 0.2702i   1.2124 + 0.8333i   0.1321 − 0.1317i  0.6874 − 0.8123i   0.1299 − 0.4106i −0.1177 + 0.4119i w21 −0.1559 +0.7442i −1.4529 − 0.2702i   0.3733 + 0.1498i   0.1321 + 0.1317i   0.4017− 0.7107i   0.1823 − 0.3822i −0.2516 + 0.3998i w22   0.1559 − 0.7442i  1.4529 + 0.2702i   0.8988 + 0.5768i −0.1321 − 0.1317i   0.6750 −1.2072i   0.2512 − 0.1287i −0.1177 − 0.4119i w23 −0.1559 − 0.7442i−1.4529 + 0.2702i   0.6394 + 0.3211i −0.1321 + 0.1317i   0.2558 −1.4247i   0.2598 − 0.1538i −0.2516 − 0.3998i w24   1.0649 + 0.2069i  0.1272 − 0.1353i   1.4730 − 0.3019i   1.4516 − 0.2578i   1.1306 +0.8649i −0.2994 − 1.4627i   0.1559 + 0.7442i w25 −1.0649 + 0.2069i−0.1272 − 0.1353i   0.1419 − 0.1122i   1.4516 + 0.2578i   0.1385 +0.7199i −0.8421 − 1.2279i   0.4328 + 0.5954i w26   1.0649 − 0.2069i  0.1272 + 0.1353i   1.0895 − 0.2172i −1.4516 − 0.2578i   1.4516 +0.2578i −1.4625 − 0.2831i   0.1559 − 0.7442i w27 −1.0649 − 0.2069i−0.1272 + 0.1353i   0.7863 − 0.1337i −1.4516 + 0.2578i   0.1689 +1.0567i −1.2488 − 0.8125i   0.4328 − 0.5954i w28   0.7408 + 0.1355i  0.1686 − 1.0718i   1.2124 − 0.8333i   0.1689 − 1.0567i   0.6874 +0.8123i −0.1299 − 0.4106i −0.1559 + 0.7442i w29 −0.7408 + 0.1355i−0.1686 − 1.0718i   0.3733 − 0.1498i   0.1689 + 1.0567i   0.4017 +0.7107i −0.1823 − 0.3822i −0.4328 + 0.5954i w30   0.7408 − 0.1355i  0.1686 + 1.0718i   0.8988 − 0.5768i −0.1689 − 1.0567i   0.6750 +1.2072i −0.2512 − 0.1287i −0.1559 − 0.7442i w31 −0.7408 − 0.1355i−0.1686 + 1.0718i   0.6394 − 0.3211i −0.1689 + 1.0567i   0.2558 +1.4247i −0.2598 − 0.1538i −0.4328 − 0.5954i w32   0.8133 + 1.2150i  0.6825 − 0.4329i −0.2775 + 1.4188i   0.6966 − 0.4427i −1.0161 −0.4912i   0.2187 + 1.0467i   1.4656 + 0.2931i w33 −0.8133 + 1.2150i−0.6825 − 0.4329i −0.1138 + 0.3999i   0.6966 + 0.4427i −0.1287 − 0.4061i  0.6179 + 0.8606i   1.2278 + 0.8230i w34   0.8133 − 1.2150i   0.6825 +0.4329i −0.2177 + 1.0243i −0.6966 − 0.4427i −1.0441 − 0.1581i   1.0296 +0.1935i   1.4656 − 0.2931i w35 −0.8133 − 1.2150i −0.6825 + 0.4329i−0.1487 + 0.7260i −0.6966 + 0.4427i −0.1321 − 0.1317i   0.8910 + 0.5575i  1.2278 − 0.8230i w36   0.2516 + 0.3998i   0.6913 − 0.8132i −0.7921 +1.2096i   0.4025 − 0.4142i −0.6966 − 0.4427i   0.1749 + 0.7440i−1.4656 + 0.2931i w37 −0.2516 + 0.3998i −0.6913 − 0.8132i −0.2891 +0.3910i   0.4025 + 0.4142i −0.4025 − 0.4142i   0.4168 + 0.6160i−1.2278 + 0.8230i w38  0.2516 − 0.3998i   0.6913 + 0.8132i −0.6056 +0.8481i −0.4025 − 0.4142i −0.6995 − 0.1411i   0.6759 + 0.1488i −1.4656 −0.2931i w39 −0.2516 − 0.3998i −0.6913 + 0.8132i −0.4397 + 0.5853i−0.4025 + 0.4142i −0.4035 − 0.1354i   0.6111 + 0.3494i −1.2278 − 0.8230iw40   1.2278 + 0.8230i   0.3948 − 0.4179i −0.2775 − 1.4188i   0.6874 −0.8123i −1.0161 + 0.4912i −0.2187 + 1.0467i   1.0649 + 0.2069i w41−1.2278 + 0.8230i −0.3948 − 0.4179i −0.1138 − 0.3999i   0.6874 + 0.8123i−0.1287 + 0.4061i −0.6179 + 0.8606i   0.8971 + 0.5677i w42   1.2278 −0.8230i   0.3948 + 0.4179i −0.2177 − 1.0243i −0.6874 − 0.8123i −1.0441 +0.1581i −1.0296 + 0.1935i   1.0649 − 0.2069i w43 −1.2278 − 0.8230i−0.3948 + 0.4179i −0.1487 − 0.7260i −0.6874 + 0.8123i −0.1321 + 0.1317i−0.8910 + 0.5575i   0.8971 − 0.5677i w44  0.3325 + 0.1582i   0.4018 −0.7177i −0.7921 − 1.2096i   0.4017 − 0.7107i −0.6966 + 0.4427i −0.1749 +0.7440i −1.0649 + 0.2069i w45 −0.3325 + 0.1582i −0.4018 − 0.7177i−0.2891 − 0.3910i   0.4017 + 0.7107i −0.4025 + 0.4142i −0.4168 + 0.6160i−0.8971 + 0.5677i w46   0.3325 − 0.1582i   0.4018 + 0.7177i −0.6056 −0.8481i −0.4017 − 0.7107i −0.6995 + 0.1411i −0.6759 + 0.1488i −1.0649 −0.2069i w47 −0.3325 − 0.1582i −0.4018 + 0.7177i −0.4397 − 0.5853i−0.4017 + 0.7107i −0.4035 + 0.1354i −0.6111 + 0.3494i −0.8971 − 0.5677iw48   0.6145 + 0.8494i   0.6796 − 0.1340i −1.4730 + 0.3019i   0.6995 −0.1411i −1.1306 − 0.8649i   0.2187 − 1.0467i   0.1678 + 0.1166i w49−0.6145 + 0.8494i −0.6796 − 0.1340i −0.1419 + 0.1122i   0.6995 + 0.1411i−0.1385 − 0.7199i   0.6179 − 0.8606i   0.3325 + 0.1582i w50   0.6145 −0.8494i   0.6796 + 0.1340i −1.0895 + 0.2172i −0.6995 − 0.1411i −1.4516 −0.2578i   1.0296 − 0.1935i   0.1678 − 0.1166i w51 −0.6145 − 0.8494i−0.6796 + 0.1340i −0.7863 + 0.1337i −0.6995 + 0.1411i −0.1689 − 1.0567i  0.8910 − 0.5575i   0.3325 − 0.1582i w52  0.4328 + 0.5954i   0.7097 −1.2125i −1.2124 + 0.8333i   0.4035 − 0.1354i −0.6874 − 0.8123i   0.1749− 0.7440i −0.1678 + 0.1166i w53 −0.4328 + 0.5954i −0.7097 − 1.2125i−0.3733 + 0.1498i   0.4035 + 0.1354i −0.4017 − 0.7107i   0.4168 −0.6160i −0.3325 + 0.1582i w54   0.4328 − 0.5954i   0.7097 + 1.2125i−0.8988 + 0.5768i −0.4035 − 0.1354i −0.6750 − 1.2072i   0.6759 − 0.1488i−0.1678 − 0.1166i w55 −0.4328 − 0.5954i −0.7097 + 1.2125i −0.6394 +0.3211i −0.4035 + 0.1354i −0.2558 − 1.4247i   0.6111 − 0.3494i −0.3325 −0.1582i w56   0.8971 + 0.5677i   0.3877 − 0.1359i −1.4730 − 0.3019i  0.6750 − 1.2072i −1.1306 + 0.8649i −0.2187 − 1.0467i   0.7408 +0.1355i w57 −0.8971 + 0.5677i −0.3877 − 0.1359i −0.1419 − 0.1122i  0.6750 + 1.2072i −0.1385 + 0.7199i −0.6179 − 0.8606i   0.6200 +0.3227i w58   0.8971 − 0.5677i   0.3877 + 0.1359i −1.0895 − 0.2172i−0.6750 − 1.2072i −1.4516 + 0.2578i −1.0296 − 0.1935i   0.7408 − 0.1355iw59 −0.8971 − 0.5677i −0.3877 + 0.1359i −0.7863 − 0.1337i −0.6750 +1.2072i −0.1689 + 1.0567i −0.8910 − 0.5575i   0.6200 − 0.3227i w60  0.6200 + 0.3227i   0.2732 − 1.4375i −1.2124 − 0.8333i   0.2558 −1.4247i −0.6874 + 0.8123i −0.1749 − 0.7440i −0.7408 + 0.1355i w61−0.6200 + 0.3227i −0.2732 − 1.4375i −0.3733 − 0.1498i   0.2558 + 1.4247i−0.4017 + 0.7107i −0.4168 − 0.6160i −0.6200 + 0.3227i w62   0.6200 −0.3227i   0.2732 + 1.4375i −0.8988 − 0.5768i −0.2558 − 1.4247i −0.6750 +1.2072i −0.6759 − 0.1488i −0.7408 − 0.1355i w63 −0.6200 − 0.3227i−0.2732 + 1.4375i −0.6394 − 0.3211i −0.2558 + 1.4247i −0.2558 + 1.4247i−0.6111 − 0.3494i −0.6200 − 0.3227i

E5) 128-QAM 2D NUC w/code rate 135/180 140/180 w0   1.1438 + 0.4323i  1.1021 + 0.4008i w1   0.9006 + 0.4173i   0.8567 + 0.4063i w2  0.5578 + 0.4216i   0.5499 + 0.4932i w3   0.6797 + 0.3850i   0.6209 +0.3666i w4   1.1837 + 0.8001i   0.1083 + 0.5383i w5   0.9008 + 0.6972i  0.1008 + 0.3768i w6   0.5354 + 0.6351i   0.3317 + 0.5081i w7  0.6549 + 0.7546i   0.3204 + 0.3654i w8   1.1598 + 0.1437i   1.0961 +0.1322i w9   0.9102 + 0.1324i   0.8513 + 0.1298i w10   0.5755 + 0.1222i  0.5343 + 0.0862i w11   0.6787 + 0.1401i   0.6260 + 0.1694i w12  1.4855 + 0.1793i   0.1082 + 0.0775i w13   1.5681 + 0.5636i   0.1037 +0.2076i w14   0.9727 + 1.2109i   0.3325 + 0.0910i w15   0.7102 + 1.0171i  0.3189 + 0.2137i w16   0.1045 + 0.4578i   1.1805 + 0.7084i w17  0.1067 + 0.3515i   0.8930 + 0.6876i w18   0.3411 + 0.4189i   0.5622 +0.6728i w19   0.3088 + 0.3456i   0.6763 + 0.8371i w20   0.1121 + 0.6749i  0.1170 + 0.7305i w21   0.1179 + 0.8955i   0.1259 + 0.9453i w22  0.3342 + 0.6628i   0.3446 + 0.7129i w23   0.3585 + 0.8699i   0.3857 +0.9233i w24   0.1075 + 0.0988i   1.4038 + 0.1588i w25   0.1136 + 0.1402i  1.5205 + 0.4861i w26   0.3700 + 0.1151i   1.0743 + 1.1040i w27  0.3294 + 0.1448i   0.7513 + 1.0912i w28   0.1933 + 1.4991i   0.1870 +1.5043i w29   0.1440 + 1.1691i   0.1454 + 1.1973i w30   0.5835 + 1.4250i  0.5736 + 1.4545i w31   0.4344 + 1.1140i   0.4433 + 1.1606i w32−1.1438 + 0.4323i   1.1021 − 0.4008i w33 −0.9006 + 0.4173i   0.8567 −0.4063i w34 −0.5578 + 0.4216i   0.5499 − 0.4932i w35 −0.6797 + 0.3850i  0.6209 − 0.3666i w36 −1.1837 + 0.8001i   0.1083 − 0.5383i w37−0.9008 + 0.6972i   0.1008 − 0.3768i w38 −0.5354 + 0.6351i   0.3317 −0.5081i w39 −0.6549 + 0.7546i   0.3204 − 0.3654i w40 −1.1598 + 0.1437i  1.0961 − 0.1322i w41 −0.9102 + 0.1324i   0.8513 − 0.1298i w42−0.5755 + 0.1222i   0.5343 − 0.0862i w43 −0.6787 + 0.1401i   0.6260 −0.1694i w44 −1.4855 + 0.1793i   0.1082 − 0.0775i w45 −1.5681 + 0.5636i  0.1037 − 0.2076i w46 −0.9727 + 1.2109i   0.3325 − 0.0910i w47−0.7102 + 1.0171i   0.3189 − 0.2137i w48 −0.1045 + 0.4578i   1.1805 −0.7084i w49 −0.1067 + 0.3515i   0.8930 − 0.6876i w50 −0.3411 + 0.4189i  0.5622 − 0.6728i w51 −0.3088 + 0.3456i   0.6763 − 0.8371i w52−0.1121 + 0.6749i   0.1170 − 0.7305i w53 −0.1179 + 0.8955i   0.1259 −0.9453i w54 −0.3342 + 0.6628i   0.3446 − 0.7129i w55 −0.3585 + 0.8699i  0.3857 − 0.9233i w56 −0.1075 + 0.0988i   1.4038 − 0.1588i w57−0.1136 + 0.1402i   1.5205 − 0.4861i w58 −0.3700 + 0.1151i   1.0743 −1.1040i w59 −0.3294 + 0.1448i   0.7513 − 1.0912i w60 −0.1933 + 1.4991i  0.1870 − 1.5043i w61 −0.1440 + 1.1691i   0.1454 − 1.1973i w62−0.5835 + 1.4250i   0.5736 − 1.4545i w63 −0.4344 + 1.1140i   0.4433 −1.1606i w64   1.1438 − 0.4323i −1.1021 + 0.4008i w65   0.9006 − 0.4173i−0.8567 + 0.4063i w66   0.5578 − 0.4216i −0.5499 + 0.4932i w67   0.6797− 0.3850i −0.6209 + 0.3666i w68   1.1837 − 0.8001i −0.1083 + 0.5383i w69  0.9008 − 0.6972i −0.1008 + 0.3768i w70   0.5354 − 0.6351i −0.3317 +0.5081i w71   0.6549 − 0.7546i −0.3204 + 0.3654i w72   1.1598 − 0.1437i−1.0961 + 0.1322i w73   0.9102 − 0.1324i −0.8513 + 0.1298i w74   0.5755− 0.1222i −0.5343 + 0.0862i w75   0.6787 − 0.1401i −0.6260 + 0.1694i w76  1.4855 − 0.1793i −0.1082 + 0.0775i w77   1.5681 − 0.5636i −0.1037 +0.2076i w78   0.9727 − 1.2109i −0.3325 + 0.0910i w79   0.7102 − 1.0171i−0.3189 + 0.2137i w80   0.1045 − 0.4578i −1.1805 + 0.7084i w81   0.1067− 0.3515i −0.8930 + 0.6876i w82   0.3411 − 0.4189i −0.5622 + 0.6728i w83  0.3088 − 0.3456i −0.6763 + 0.8371i w84   0.1121 − 0.6749i −0.1170 +0.7305i w85   0.1179 − 0.8955i −0.1259 + 0.9453i w86   0.3342 − 0.6628i−0.3446 + 0.7129i w87   0.3585 − 0.8699i −0.3857 + 0.9233i w88   0.1075− 0.0988i −1.4038 + 0.1588i w89   0.1136 − 0.1402i −1.5205 + 0.4861i w90  0.3700 − 0.1151i −1.0743 + 1.1040i w91   0.3294 − 0.1448i −0.7513 +1.0912i w92   0.1933 − 1.4991i −0.1870 + 1.5043i w93   0.1440 − 1.1691i−0.1454 + 1.1973i w94   0.5835 − 1.4250i −0.5736 + 1.4545i w95   0.4344− 1.1140i −0.4433 + 1.1606i w96 −1.1438 − 0.4323i −1.1021 − 0.4008i w97−0.9006 − 0.4173i −0.8567 − 0.4063i w98 −0.5578 − 0.4216i −0.5499 −0.4932i w99 −0.6797 − 0.3850i −0.6209 − 0.3666i w100 −1.1837 − 0.8001i−0.1083 − 0.5383i w101 −0.9008 − 0.6972i −0.1008 − 0.3768i w102 −0.5354− 0.6351i −0.3317 − 0.5081i w103 −0.6549 − 0.7546i −0.3204 − 0.3654iw104 −1.1598 − 0.1437i −1.0961 − 0.1322i w105 −0.9102 − 0.1324i −0.8513− 0.1298i w106 −0.5755 − 0.1222i −0.5343 − 0.0862i w107 −0.6787 −0.1401i −0.6260 − 0.1694i w108 −1.4855 − 0.1793i −0.1082 − 0.0775i w109−1.5681 − 0.5636i −0.1037 − 0.2076i w110 −0.9727 − 1.2109i −0.3325 −0.0910i w111 −0.7102 − 1.0171i −0.3189 − 0.2137i w112 −0.1045 − 0.4578i−1.1805 − 0.7084i w113 −0.1067 − 0.3515i −0.8930 − 0.6876i w114 −0.3411− 0.4189i −0.5622 − 0.6728i w115 −0.3088 − 0.3456i −0.6763 − 0.8371iw116 −0.1121 − 0.6749i −0.1170 − 0.7305i w117 −0.1179 − 0.8955i −0.1259− 0.9453i w118 −0.3342 − 0.6628i −0.3446 − 0.7129i w119 −0.3585 −0.8699i −0.3857 − 0.9233i w120 −0.1075 − 0.0988i −1.4038 − 0.1588i w121−0.1136 − 0.1402i −1.5205 − 0.4861i w122 −0.3700 − 0.1151i −1.0743 −1.1040i w123 −0.3294 − 0.1448i −0.7513 − 1.0912i w124 −0.1933 − 1.4991i−0.1870 − 15043i w125 −0.1440 − 1.1691i −0.1454 − 1.1973i w126 −0.5835 −1.4250i −0.5736 − 1.4545i w127 −0.4344 − 1.1140i −0.4433 − 1.1606i

E6) 256-QAM 2D NUC w/code rate 20/30 22/30 w0   1.6350 + 0.1593i  1.5977 + 0.1526i w1   1.5776 + 0.4735i   1.3187 + 0.1269i w2  0.9430 + 0.1100i −1.5977 + 0.1526i w3   0.9069 + 0.2829i −1.3187 +0.1269i w4   0.3237 + 0.0849i   0.2574 + 0.0733i w5   0.3228 + 0.0867i  0.4496 + 0.0807i w6   0.7502 + 0.1138i −0.2574 + 0.0733i w7   0.7325 +0.2088i −0.4496 + 0.0807i w8   0.1658 + 1.6747i   1.5977 − 0.1526i w9  0.4907 + 1.6084i   1.3187 − 0.1269i w10   0.1088 + 0.9530i −1.5977 −0.1526i w11   0.2464 + 0.9270i −1.3187 − 0.1269i w12   0.0872 + 0.1390i  0.2574 − 0.0733i w13   0.0871 + 0.1392i   0.4496 − 0.0807i w14  0.1091 + 0.7656i −0.2574 − 0.0733i w15   0.1699 + 0.7537i −0.4496 −0.0807i w16 −1.6350 + 0.1593i   0.9269 + 0.0943i w17 −1.5776 + 0.4735i  1.1024 + 0.1086i w18 −0.9430 + 0.1100i −0.9269 + 0.0943i w19 −0.9069 +0.2829i −1.1024 + 0.1086i w20 −0.3237 + 0.0849i   0.7663 + 0.0867i w21−0.3228 + 0.0867i   0.6115 + 0.0871i w22 −0.7502 + 0.1138i −0.7663 +0.0867i w23 −0.7325 + 0.2088i −0.6115 + 0.0871i w24 −0.1658 + 1.6747i  0.9269 − 0.0943i w25 −0.4907 + 1.6084i   1.1024 − 0.1086i w26−0.1088 + 0.9530i −0.9269 − 0.0943i w27 −0.2464 + 0.9270i −1.1024 −0.1086i w28 −0.0872 + 0.1390i   0.7663 − 0.0867i w29 −0.0871 + 0.1392i  0.6115 − 0.0871i w30 −0.1091 + 0.7656i −0.7663 − 0.0867i w31 −0.1699 +0.7537i −0.6115 − 0.0871i w32   1.3225 + 0.1320i   1.2701 + 1.0139i w33  1.2742 + 0.3922i   1.0525 + 0.8406i w34   1.0854 + 0.1139i −1.2701 +1.0139i w35   1.0441 + 0.3296i −1.0525 + 0.8406i w36   0.4582 + 0.1123i  0.2487 + 0.1978i w37   0.4545 + 0.1251i   0.3523 + 0.2915i w38  0.6473 + 0.1138i −0.2487 + 0.1978i w39   0.6339 + 0.1702i −0.3523 +0.2915i w40   0.1322 + 1.3631i   1.2701 − 1.0139i w41   0.3929 + 1.3102i  1.0525 − 0.8406i w42   0.1124 + 1.1327i −1.2701 − 1.0139i w43  0.3160 + 1.0913i −1.0525 − 0.8406i w44   0.0928 + 0.3970i   0.2487 −0.1978i w45   0.0937 + 0.3973i   0.3523 − 0.2915i w46   0.1054 + 0.5979i−0.2487 − 0.1978i w47   0.1230 + 0.5949i −0.3523 − 0.2915i w48 −1.3225 +0.1320i   0.7359 + 0.6043i w49 −1.2742 + 0.3922i   0.8807 + 0.7105i w50−1.0854 + 0.1139i −0.7359 + 0.6043i w51 −1.0441 + 0.3296i −0.8807 +0.7105i w52 −0.4582 + 0.1123i   0.6017 + 0.5019i w53 −0.4545 + 0.1251i  0.4747 + 0.3996i w54 −0.6473 + 0.1138i −0.6017 + 0.5019i w55 −0.6339 +0.1702i −0.4747 + 0.3996i w56 −0.1322 + 1.3631i   0.7359 − 0.6043i w57−0.3929 + 1.3102i   0.8807 − 0.7105i w58 −0.1124 + 1.1327i −0.7359 −0.6043i w59 −0.3160 + 1.0913i −0.8807 − 0.7105i w60 −0.0928 + 0.3970i  0.6017 − 0.5019i w61 −0.0937 + 0.3973i   0.4747 − 0.3996i w62−0.1054 + 0.5979i −0.6017 − 0.5019i w63 −0.1230 + 0.5949i −0.4747 −0.3996i w64   1.6350 − 0.1593i   1.5441 + 0.4545i w65   1.5776 − 0.4735i  1.2750 + 0.3775i w66   0.9430 − 0.1100i −1.5441 + 0.4545i w67   0.9069− 0.2829i −1.2750 + 0.3775i w68   0.3237 − 0.0849i   0.2586 + 0.0752iw69   0.3228 − 0.0867i   0.4435 + 0.1065i w70   0.7502 − 0.1138i−0.2586 + 0.0752i w71   0.7325 − 0.2088i −0.4435 + 0.1065i w72   0.1658− 1.6747i   1.5441 − 0.4545i w73   0.4907 − 1.6084i   1.2750 − 0.3775iw74   0.1088 − 0.9530i −1.5441 − 0.4545i w75   0.2464 − 0.9270i −1.2750− 0.3775i w76   0.0872 − 0.1390i −0.2586 − 0.0752i w77   0.0871 −0.1392i   0.4435 − 0.1065i w78   0.1091 − 0.7656i −0.2586 − 0.0752i w79  0.1699 − 0.7537i −0.4435 − 0.1065i w80 −1.6350 − 0.1593i   0.8925 +0.2771i w81 −1.5776 − 0.4735i   1.0649 + 0.3219i w82 −0.9430 − 0.1100i−0.8925 + 0.2771i w83 −0.9069 − 0.2829i −1.0649 + 0.3219i w84 −0.3237 −0.0849i   0.7362 + 0.2279i w85 −0.3228 − 0.0867i   0.5936 + 0.1699i w86−0.7502 − 0.1138i −0.7362 + 0.2279i w87 −0.7325 − 0.2088i −0.5936 +0.1699i w88 −0.1658 − 1.6747i   0.8925 − 0.2771i w89 −0.4907 − 1.6084i  1.0649 − 0.3219i w90 −0.1088 − 0.9530i −0.8925 − 0.2771i w91 −0.2464 −0.9270i −1.0649 − 0.3219i w92 −0.0872 − 0.1390i   0.7362 − 0.2279i w93−0.0871 − 0.1392i   0.5936 − 0.1699i w94 −0.1091 − 0.7656i −0.7362 −0.2279i w95 −0.1699 − 0.7537i −0.5936 − 0.1699i w96   1.3225 − 0.1320i  1.4352 + 0.7452i w97   1.2742 − 0.3922i   1.1866 + 0.6182i w98  1.0854 − 0.1139i −1.4352 + 0.7452i w99   1.0441 − 0.3296i −1.1866 +0.6182i w100   0.4582 − 0.1123i   0.2523 + 0.1944i w101   0.4545 −0.1251i   0.3695 + 0.2695i w102   0.6473 − 0.1138i −0.2523 + 0.1944iw103   0.6339 − 0.1702i −0.3695 + 0.2695i w104   0.1322 − 1.3631i  1.4352 − 0.7452i w105   0.3929 − 1.3102i   1.1866 − 0.6182i w106  0.1124 − 1.1327i −1.4352 − 0.7452i w107   0.3160 − 1.0913i −1.1866 −0.6182i w108   0.0928 − 0.3970i   0.2523 − 0.1944i w109   0.0937 −0.3973i   0.3695 − 0.2695i w110   0.1054 − 0.5979i −0.2523 − 0.1944iw111   0.1230 − 0.5949i −0.3695 − 0.2695i w112 −1.3225 − 0.1320i  0.8273 + 0.4493i w113 −1.2742 − 0.3922i   0.9911 + 0.5243i w114−1.0854 − 0.1139i −0.8273 + 0.4493i w115 −1.0441 − 0.3296i −0.9911 +0.5243i w116 −0.4582 − 0.1123i   0.6708 + 0.3859i w117 −0.4545 − 0.1251i  0.5197 + 0.3331i w118 −0.6473 − 0.1138i −0.6708 + 0.3859i w119 −0.6339− 0.1702i −0.5197 + 0.3331i w120 −0.1322 − 1.3631i   0.8273 − 0.4493iw121 −0.3929 − 1.3102i   0.9911 − 0.5243i w122 −0.1124 − 1.1327i −0.8273− 0.4493i w123 −0.3160 − 1.0913i −0.9911 − 0.5243i w124 −0.0928 −0.3970i   0.6708 − 0.3859i w125 −0.0937 − 0.3973i   0.5197 − 0.3331iw126 −0.1054 − 0.5979i −0.6708 − 0.3859i w127 −0.1230 − 0.5949i −0.5197− 0.3331i w128   1.2901 + 1.0495i   0.1646 + 1.6329i w129   1.4625 +0.7740i   0.1379 + 1.3595i w130   0.7273 + 0.6160i −0.1646 + 1.6329iw131   0.8177 + 0.4841i −0.1379 + 1.3595i w132   0.2844 + 0.1296i  0.0736 + 0.0898i w133   0.2853 + 0.1309i   0.0742 + 0.5054i w134  0.5902 + 0.4857i −0.0736 + 0.0898i w135   0.6355 + 0.4185i −0.0742 +0.5054i w136   1.0646 + 1.2876i   0.1646 − 1.6329i w137   0.7949 +1.4772i   0.1379 − 1.3595i w138   0.5707 + 0.7662i −0.1646 − 1.6329iw139   0.4490 + 0.8461i −0.1379 − 1.3595i w140   0.1053 + 0.1494i  0.0736 − 0.0898i w141   0.1052 + 0.1495i   0.0742 − 0.5054i w142  0.4294 + 0.6363i −0.0736 − 0.0898i w143   0.3744 + 0.6744i −0.0742 −0.5054i w144 −1.2901 + 1.0495i   0.0992 + 0.9847i w145 −1.4625 + 0.7740i  0.1170 + 1.1517i w146 −0.7273 + 0.6160i −0.0992 + 0.9847i w147−0.8177 + 0.4841i −0.1170 + 1.1517i w148 −0.2844 + 0.1296i   0.0894 +0.8287i w149 −0.2853 + 0.1309i   0.0889 + 0.6739i w150 −0.5902 + 0.4857i−0.0894 + 0.8287i w151 −0.6355 + 0.4185i −0.0889 + 0.6739i w152−1.0646 + 1.2876i   0.0992 − 0.9847i w153 −0.7949 + 1.4772i   0.1170 −1.1517i w154 −0.5707 + 0.7662i −0.0992 − 0.9847i w155 −0.4490 + 0.8461i−0.1170 − 1.1517i w156 −0.1053 + 0.1494i   0.0894 − 0.8287i w157−0.1052 + 0.1495i   0.0889 − 0.6739i w158 −0.4294 + 0.6363i −0.0894 −0.8287i w159 −0.3744 + 0.6744i −0.0889 − 0.6739i w160   1.0382 + 0.8623i  1.0516 + 1.2481i w161   1.1794 + 0.6376i   0.8742 + 1.0355i w162  0.8504 + 0.7217i −1.0516 + 1.2481i w163   0.9638 + 0.5407i −0.8742 +1.0355i w164   0.3734 + 0.2560i   0.0970 + 0.2450i w165   0.3799 +0.2517i   0.1959 + 0.4045i w166   0.4968 + 0.3947i −0.0970 + 0.2450iw167   0.5231 + 0.3644i −0.1959 + 0.4045i w168   0.8555 + 1.0542i  1.0516 − 1.2481i w169   0.6363 + 1.2064i   0.8742 − 1.0355i w170  0.6961 + 0.8850i −1.0516 − 1.2481i w171   0.5229 + 1.0037i −0.8742 −1.0355i w172   0.1938 + 0.3621i   0.0970 − 0.2450i w173   0.1909 +0.3627i   0.1959 − 0.4045i w174   0.3224 + 0.5236i −0.0970 − 0.2450iw175   0.3016 + 0.5347i −0.1959 − 0.4045i w176 −1.0382 + 0.8623i  0.6150 + 0.7441i w177 −1.1794 + 0.6376i   0.7345 + 0.8743i w178−0.8504 + 0.7217i −0.6150 + 0.7441i w179 −0.9638 + 0.5407i −0.7345 +0.8743i w180 −0.3734 + 0.2560i   0.4932 + 0.6301i w181 −0.3799 + 0.2517i  0.3620 + 0.5258i w182 −0.4968 + 0.3947i −0.4932 + 0.6301i w183−0.5231 + 0.3644i −0.3620 + 0.5258i w184 −0.8555 + 1.0542i   0.6150 −0.7441i w185 −0.6363 + 1.2064i   0.7345 − 0.8743i w186 −0.6961 + 0.8850i−0.6150 − 0.7441i w187 −0.5229 + 1.0037i −0.7345 − 0.8743i w188−0.1938 + 0.3621i   0.4932 − 0.6301i w189 −0.1909 + 0.3627i   0.3620 −0.5258i w190 −0.3224 + 0.5236i −0.4932 − 0.6301i w191 −0.3016 + 0.5347i−0.3620 − 0.5258i w192   1.2901 − 1.0495i   0.4866 + 1.5660i w193  1.4625 − 0.7740i   0.4068 + 1.3027i w194   0.7273 − 0.6160i −0.4866 +1.5660i w195   0.8177 − 0.4841i −0.4068 + 1.3027i w196   0.2844 −0.1296i   0.0732 + 0.0899i w197   0.2853 − 0.1309i   0.0877 + 0.4997iw198   0.5902 − 0.4857i −0.0732 + 0.0899i w199   0.6355 − 0.4185i−0.0877 + 0.4997i w200   1.0646 − 1.2876i   0.4866 − 1.5660i w201  0.7949 − 1.4772i   0.4068 − 1.3027i w202   0.5707 − 0.7662i −0.4866 −1.5660i w203   0.4490 − 0.8461i −0.4068 − 1.3027i w204   0.1053 −0.1494i   0.0732 − 0.0899i w205   0.1052 − 0.1495i   0.0877 − 0.4997iw206   0.4294 − 0.6363i −0.0732 − 0.0899i w207   0.3744 − 0.6744i−0.0877 − 0.4997i w208 −1.2901 − 1.0495i   0.2927 + 0.9409i w209 −1.4625− 0.7740i   0.3446 + 1.1023i w210 −0.7273 − 0.6160i −0.2927 + 0.9409iw211 −0.8177 − 0.4841i −0.3446 + 1.1023i w212 −0.2844 − 0.1296i  0.2350 + 0.7945i w213 −0.2853 − 0.1309i   0.1670 + 0.6529i w214−0.5902 − 0.4857i −0.2350 + 0.7945i w215 −0.6355 − 0.4185i −0.1670 +0.6529i w216 −1.0646 − 1.2876i   0.2927 − 0.9409i w217 −0.7949 − 1.4772i  0.3446 − 1.1023i w218 −0.5707 − 0.7662i −0.2927 − 0.9409i w219 −0.4490− 0.8461i −0.3446 − 1.1023i w220 −0.1053 − 0.1494i   0.2350 − 0.7945iw221 −0.1052 − 0.1495i   0.1670 − 0.6529i w222 −0.4294 − 0.6363i −0.2350− 0.7945i w223 −0.3744 − 0.6744i −0.1670 − 0.6529i w224   1.0382 −0.8623i   0.7867 + 1.4356i w225   1.1794 − 0.6376i   0.6561 + 1.1927iw226   0.8504 − 0.7217i −0.7867 + 1.4356i w227   0.9638 − 0.5407i−0.6561 + 1.1927i w228   0.3734 − 0.2560i   0.0947 + 0.2451i w229  0.3799 − 0.2517i   0.1865 + 0.4121i w230   0.4968 − 0.3947i −0.0947 +0.2451i w231   0.5231 − 0.3644i −0.1865 + 0.4121i w232   0.8555 −1.0542i   0.7867 − 1.4356i w233   0.6363 − 1.2064i   0.6561 − 1.1927iw234   0.6961 − 0.8850i −0.7867 − 1.4356i w235   0.5229 − 1.0037i−0.6561 − 1.1927i w236   0.1938 − 0.3621i   0.0947 − 0.2451i w237  0.1909 − 0.3627i   0.1865 − 0.4121i w238   0.3224 − 0.5236i −0.0947 −0.2451i w239   0.3016 − 0.5347i −0.1865 − 0.4121i w240 −1.0382 − 0.8623i−0.4677 + 0.8579i w241 −1.1794 − 0.6376i   0.5537 + 1.0081i w242 −0.8504− 0.7217i −0.4677 + 0.8579i w243 −0.9638 − 0.5407i −0.5537 + 1.0081iw244 −0.3734 − 0.2560i   0.3893 + 0.7143i w245 −0.3799 − 0.2517i  0.3110 + 0.5686i w246 −0.4968 − 0.3947i −0.3893 + 0.7143i w247 −0.5231− 0.3644i −0.3110 + 0.5686i w248 −0.8555 − 1.0542i   0.4677 − 0.8579iw249 −0.6363 − 1.2064i   0.5537 − 1.0081i w250 −0.6961 − 0.8850i −0.4677− 0.8579i w251 −0.5229 − 1.0037i −0.5537 − 1.0081i w252 −0.1938 −0.3621i   0.3893 − 0.7143i w253 −0.1909 − 0.3627i   0.3110 − 0.5686iw254 −0.3224 − 0.5236i −0.3893 − 0.7143i w255 −0.3016 − 0.5347i −0.3110− 0.5686i

Constellations from group E can preferably be used for coding andmodulation in accordance with the DVB-S2 standard or its extensionDVB-Sx. Special care has been taken such that the constellations can beused together with LDPC codes and bit interleaver settings from theDVB-Sx baseline system.

Whenever constellations are proposed for LDPC codes not yet part of theDVB-Sx baseline, like e.g. codes of code rate “x/30”, the constellationsare optimized such that they allow for a DVB-S2-like bit interleaver.This means, the same interleaving rule can be applied as used in DVBS2for 64 k LDPC codes (except for code rate 3/5 from S2): the blockinterleaver is filled column-wise, and read out row-wise, each row readfrom left to right. In terms of bit interleaver patterns, as discussedin the DVB-Sx baseline, this would correspond to bit interleaverpatterns [0,1,2, . . . M−1], in which M is the number of bits/QAMsymbol, e.g., M=3 for a constellation with 8 points, M=4, for 16 points,and so on.

It should be noted that the code rates for which the constellations ofgroups A, B, C, D and E have been optimized are the code rates of theLDPC encoder. However, the total code rates may actually be smaller dueto the use of an additional BCH encoder.

FIG. 13 shows a diagram for a constellation from group E for 8-NUC forcode rate R=100/180 or 104/180. The corresponding constellation pointvector isw=(−0.2330,−0.5414−0.8712i,−0.5414+0.8712i,−1.3570,0.2330,0.5414−0.8712i,0.5414+0.8712i,1.3570).

FIG. 14 shows a diagram illustrating the shaping gain for the aboveexplained non-uniform 8-ary QAM constellations which may be used for astandard relating to digital video broadcasting via satellite. Thecurves were obtained applying the following conditions: AWGN channel,bit error rate and frame error rates (BER, FER) after 50 LDPCiterations, and outer BCH decoding; 64 k LDPC codes were used (codelength=64800). The diagram includes curves showing the BER (solid lines)and the FER (dashed lines). Further, there are curves for conventionalconstellations as used in known standards using baseline constellationsand curves for the above explained NUCs for 8-ary QAMs. It can be seenthat the same BER or FER can be achieved at lower SNR. The best resultis obtained for code rate 13/18: the proposed constellation works at0.55 dB lower SNR than the conventional constellation. The curves arealmost shifted in parallel to the left with the optimized constellationby this offset. For code rate 25/36 a 0.5 dB gain is achieved.

FIGS. 15AA-15CB show examples of LDPC codes for which the disclosedconstellations have been optimized. The code length and the code rateare given for the respective codes.

It should be noted that the present disclosure is to be understood suchthat the disclosure includes embodiments of coding and modulationapparatus for which less groups of tables of constellations areavailable for selection and/or use of a constellation, for which smallertables of constellations are available for selection and/or use of aconstellation, for which tables including constellations for less coderates and/or less values of M are available for selection and/or use ofa constellation, and/or for which only selected (single) constellationsfrom among all the disclosed constellations are available for selectionand/or use of a constellation.

Obviously, numerous modifications and variations of the presentdisclosure are possible in light of the above teachings. It is thereforeto be understood that within the scope of the appended claims, thedisclosure may be practiced otherwise than as specifically describedherein (e.g., if the NUC position vectors are rounded to a smallernumber of digits).

In the claims, the word “comprising” does not exclude other elements orsteps, and the indefinite article “a” or “an” does not exclude aplurality. A single element or other unit may fulfill the functions ofseveral items recited in the claims. The mere fact that certain measuresare recited in mutually different dependent claims does not indicatethat a combination of these measures cannot be used to advantage.

In so far as embodiments of the disclosure have been described as beingimplemented, at least in part, by software-controlled data processingapparatus, it will be appreciated that a non-transitory machine-readablemedium carrying such software, such as an optical disk, a magnetic disk,semiconductor memory or the like, is also considered to represent anembodiment of the present disclosure. Further, such a software may alsobe distributed in other forms, such as via the Internet or other wiredor wireless telecommunication systems.

A circuit that may be used for implementing one or more of the elementsof the claimed apparatus is a structural assemblage of electroniccomponents including conventional circuit elements, integrated circuitsincluding application specific integrated circuits, standard integratedcircuits, application specific standard products, and field programmablegate arrays. Further a circuit includes central processing units,graphics processing units, and microprocessors which are programmed orconfigured according to software code. A circuit does not include puresoftware, although a circuit includes the above-described hardwareexecuting software.

Any reference signs in the claims should not be construed as limitingthe scope.

The invention claimed is:
 1. A receiving apparatus comprising: areceiver configured to receive at least one transmission streamincluding symbols representative of constellation values; and at leastone processor configured to demodulate the symbols into cell words andwherein each of the symbols is associated with a point of a non-uniformconstellation (NUC), wherein the NUC includes 64 points, wherein a firstquarter of the NUC includes points defined by following real andimaginary parts: 0.5656+j0.9499 0.2750+j1.0676 0.8202+j1.26510.3011+j1.4529 0.9500+j0.5641 1.0666+j0.2744 1.2657+j0.81781.4521+j0.3005 0.3004+j0.5417 0.2430+j0.5607 0.2094+j0.35790.1946+j0.3566 0.5410+j0.3002 0.5596+j0.2431 0.3569+j0.20940.3553+j0.1948.
 2. The receiving apparatus according to claim 1, whereina second quarter of the NUC includes 16 points, each being an oppositeof a complex conjugate (−conj) of a point of the first quarter; whereina third quarter of the NUC includes 16 points, each being a complexconjugate (conj) of a point of the first quarter; wherein a fourthquarter of the NUC includes 16 points, each being an opposite of a pointof the first quarter.
 3. The receiving apparatus according to claim 1,wherein the at least one transmission stream is a digital videobroadcasting stream.
 4. The receiving apparatus according to claim 1,wherein the at least one transmission stream is a terrestrial digitalvideo broadcasting stream.
 5. The receiving apparatus according to claim1, wherein the NUC is a Quadrature Amplitude Modulation (QAM)constellation.
 6. The receiving apparatus according to claim 1, whereinbits of the cell words are labeled on the NUC in accordance with a Graymapping.
 7. The receiving apparatus according to claim 1, wherein the atleast one processor is configured to decode the cell words into outputdata.
 8. A receiving method comprising: receiving at least onetransmission stream including symbols representative of constellationvalues; and demodulating the symbols into cell words, wherein each ofthe symbols is associated with a point of a non-uniform constellation(NUC), wherein the NUC includes 64 points, wherein a first quarter ofthe NUC includes points defined by following real and imaginary parts:0.5656+j0.9499 0.2750+0.0676 0.8202+j1.2651 0.3011+j1.45290.9500+j0.5641 1.0666+j0.2744 1.2657+j0.8178 1.4521+j0.30050.3004+j0.5417 0.2430+j0.5607 0.2094+j0.3579 0.1946+j0.35660.5410+j0.3002 0.5596+j0.2431 0.3569+j0.2094 0.3553+j0.1948.
 9. Thereceiving method according to claim 8, wherein a second quarter of theNUC includes 16 points, each being an opposite of a complex conjugate(−conj) of a point of the first quarter, wherein a third quarter of theNUC includes 16 points, each being a complex conjugate (conj) of a pointof the first quarter, and wherein a fourth quarter of the NUC includes16 points, each being an opposite of a point of the first quarter. 10.The receiving method according to claim 8, wherein the at least onetransmission stream is a digital video broadcasting stream.
 11. Areceiving apparatus comprising: a receiver configured to receive atleast one transmission stream including symbols representative ofconstellation values; and at least one processor configured todemodulate the symbols into cell words and wherein each of the symbolsis associated with a point of a non-uniform constellation (NUC), whereinthe NUC includes 64 points, wherein a first quarter of the NUC includespoints defined by following real and imaginary parts: 0.2920+j1.48270.2174+j1.0211 0.5702+j0.8798 1.4827+j0.2920 1.0211+j0.21740.8798+j0.5702 0.1475+j0.3040 0.1691+j0.3028 0.1871+j0.68550.3563+j0.6126 0.3040+j0.1475 0.3028+j0.1691 0.6855+j0.18710.6126+j0.3563.
 12. The receiving apparatus according to claim 11,wherein a second quarter of the NUC includes 16 points, each being anopposite of a complex conjugate (−conj) of a point of the first quarter,wherein a third quarter of the NUC includes 16 points, each being acomplex conjugate (conj) of a point of the first quarter, and wherein afourth quarter of the NUC includes 16 points, each being an opposite ofa point of the first quarter.
 13. The receiving apparatus according toclaim 11, wherein the first quarter of the NUC includes points definedby following real and imaginary parts: 0.8410+0.2563 1.2563+j0.8411. 14.The receiving apparatus according to claim 11, wherein the at least onetransmission stream is a digital video broadcasting stream.
 15. Thereceiving apparatus according to claim 11, wherein the at least onetransmission stream is a terrestrial digital video broadcasting stream.16. The receiving apparatus according to claim 11, wherein the NUC is aQuadrature Amplitude Modulation (QAM) constellation.
 17. The receivingapparatus according to claim 11, wherein bits of the cell words arelabeled on the NUC in accordance with a Gray mapping.
 18. The receivingapparatus according to claim 11, wherein the at least one processor isconfigured to decode the cell words into output data.
 19. A receivingmethod comprising: receiving at least one transmission stream includingsymbols representative of constellation values; and demodulating thesymbols into cell words, wherein each of the symbols is associated witha point of a non-uniform constellation (NUC), wherein the NUC includes64 points, and wherein a first quarter of the NUC includes pointsdefined by following real and imaginary parts: 0.2920+j 1.48270.2174+j1.0211 0.5702+0.8798 1.4827+j0.2920 1.0211+j0.21740.8798+j0.5702 0.1475+j0.3040 0.1691+j0.3028 0.1871+j0.68550.3563+j0.6126 0.3040+j0.1475 0.3028+j0.1691 0.6855+j0.18710.6126+j0.3563.
 20. The receiving method according to claim 19, whereina second quarter of the NUC includes 16 points, each being an oppositeof a complex conjugate (−conj) of a point of the first quarter, whereina third quarter of the NUC includes 16 points, each being a complexconjugate (conj) of a point of the first quarter, and wherein a fourthquarter of the NUC includes 16 points, each being an opposite of a pointof the first quarter.
 21. The receiving method according to claim 19,wherein a first quarter of the NUC includes points defined by followingreal and imaginary parts: 0.8410+j1.2563 1.2563+j0.8411.
 22. Thereceiving method according to claim 19, wherein the at least onetransmission stream is a digital video broadcasting stream.